%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SEU711^2 : TPTP v8.1.2. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n032.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 17:22:50 EDT 2023
% Result : Theorem 92.52s 92.80s
% Output : Proof 92.52s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_powerset,type,
powerset: $i > $i ).
thf(ty_binunion,type,
binunion: $i > $i > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(sP1,plain,
( sP1
<=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) )
@ ( powerset
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ~ ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $o] :
( ( in
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X4: $i] :
~ ( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) )
=> ( in @ ( binunion @ X4 @ X5 )
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) ) ) )
@ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( in @ X2
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( ( ( in
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X4: $i] :
~ ( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) )
=> ( in @ ( binunion @ X4 @ X5 )
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) ) ) )
@ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( in @ X2
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> X1 )
=> ( ( ( in
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X4: $i] :
~ ( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) )
=> ( in @ ( binunion @ X4 @ X5 )
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) ) ) )
@ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( in @ X2
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
=> X1 )
=> X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i] :
( sP1
=> ( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) )
=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i] :
( ( in
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( sP1
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ! [X1: $i,X2: $o] :
( ( in @ X1
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X4: $i] :
~ ( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) )
=> ( in @ ( binunion @ X4 @ X5 )
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) ) ) )
@ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( ( ( in @ X1
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) ) )
=> X2 )
=> ( ( ( in @ X1
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X4: $i] :
~ ( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) )
=> ( in @ ( binunion @ X4 @ X5 )
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) ) ) )
@ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
=> X2 )
=> X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ! [X1: $i,X2: $i] :
( ( ( in
@ @+[X3: $i] :
~ ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
@ X2 )
=> ( in
@ @+[X3: $i] :
~ ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
@ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: $i] :
( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ ( binunion
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ) )
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ( ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ ( binunion
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ) )
=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) )
@ ( powerset
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) )
=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ! [X1: $i] :
( ( ( in
@ @+[X2: $i] :
~ ( ( in @ X2 @ X1 )
=> ( in @ X2
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
@ X1 )
=> ( in
@ @+[X2: $i] :
~ ( ( in @ X2 @ X1 )
=> ( in @ X2
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ! [X1: $i,X2: $i] :
( ( in @ X1
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( ( in @ X2 @ X1 )
=> ( in @ X2
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
@ ( powerset
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( sP8
=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( in
@ ( binunion
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) )
@ ( powerset
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( sP19
=> sP14 ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ ( binunion
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ) )
=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ! [X1: $i] :
( ( in @ X1
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ! [X1: $i,X2: $i,X3: $o] :
( ( in @ X2
@ ( binunion
@ @+[X4: $i] :
~ ( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) )
=> ( in @ ( binunion @ X4 @ X5 )
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) ) ) )
@ X1 ) )
=> ( ( ( in @ X2
@ @+[X4: $i] :
~ ( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) )
=> ( in @ ( binunion @ X4 @ X5 )
@ ( powerset
@ @+[X6: $i] :
~ ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X6 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X6 ) )
=> ( in @ ( binunion @ X7 @ X8 ) @ ( powerset @ X6 ) ) ) ) ) ) ) ) )
=> X3 )
=> ( ( ( in @ X2 @ X1 )
=> X3 )
=> X3 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ @+[X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( in
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X3: $i] :
~ ( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) )
=> ( in @ ( binunion @ X3 @ X4 )
@ ( powerset
@ @+[X5: $i] :
~ ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X5 ) )
=> ! [X7: $i] :
( ( in @ X7 @ ( powerset @ X5 ) )
=> ( in @ ( binunion @ X6 @ X7 ) @ ( powerset @ X5 ) ) ) ) ) ) ) )
@ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) )
=> ( in @ X1
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
@ ( binunion
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ( in @ ( binunion @ X1 @ X2 )
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) ) ) )
@ @+[X1: $i] :
~ ( ( in @ X1
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) )
=> ( in
@ ( binunion
@ @+[X2: $i] :
~ ( ( in @ X2
@ ( powerset
@ @+[X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X3 ) )
=> ( in @ ( binunion @ X4 @ X5 ) @ ( powerset @ X3 ) ) ) ) ) )
=> ! [X3: $i] :
( ( in @ X3
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) )
=> ( in @ ( binunion @ X2 @ X3 )
@ ( powerset
@ @+[X4: $i] :
~ ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X4 ) )
=> ! [X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X5 @ X6 ) @ ( powerset @ X4 ) ) ) ) ) ) ) )
@ X1 )
@ ( powerset
@ @+[X2: $i] :
~ ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X2 ) )
=> ( in @ ( binunion @ X3 @ X4 ) @ ( powerset @ X2 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(def_powersetI,definition,
( powersetI
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ X1 ) )
@ ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_powersetE,definition,
( powersetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X2 @ ( powerset @ X1 ) )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_binunionEcases,definition,
( binunionEcases
= ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ ( binunion @ X1 @ X2 ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X1 )
@ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X2 )
@ X4 )
@ X4 ) ) ) ) ) ).
thf(binunionT_lem,conjecture,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( sP6
=> ( sP28
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ) ).
thf(h0,negated_conjecture,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( sP6
=> ( sP28
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[binunionT_lem]) ).
thf(h1,assumption,
sP11,
introduced(assumption,[]) ).
thf(h2,assumption,
~ ( sP6
=> ( sP28
=> ( sP19
=> ( sP1
=> sP21 ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
sP6,
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( sP28
=> ( sP19
=> ( sP1
=> sP21 ) ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
sP28,
introduced(assumption,[]) ).
thf(h6,assumption,
~ ( sP19
=> ( sP1
=> sP21 ) ),
introduced(assumption,[]) ).
thf(h7,assumption,
sP19,
introduced(assumption,[]) ).
thf(h8,assumption,
~ ( sP1
=> sP21 ),
introduced(assumption,[]) ).
thf(h9,assumption,
sP1,
introduced(assumption,[]) ).
thf(h10,assumption,
~ sP21,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP2
| sP8
| sP26 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP13
| ~ sP29
| sP2 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP16
| ~ sP26
| sP27 ),
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP20
| ~ sP8
| sP27 ),
inference(prop_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP24
| sP20 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP12
| sP16 ),
inference(all_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP3
| sP13 ),
inference(all_rule,[status(thm)],]) ).
thf(8,plain,
( ~ sP22
| ~ sP19
| sP14 ),
inference(prop_rule,[status(thm)],]) ).
thf(9,plain,
( ~ sP9
| ~ sP1
| sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( ~ sP10
| sP3 ),
inference(all_rule,[status(thm)],]) ).
thf(11,plain,
( ~ sP14
| sP24 ),
inference(choice_rule,[status(thm)],]) ).
thf(12,plain,
( ~ sP7
| sP22 ),
inference(all_rule,[status(thm)],]) ).
thf(13,plain,
( ~ sP5
| sP12 ),
inference(choice_rule,[status(thm)],]) ).
thf(14,plain,
( ~ sP4
| sP9 ),
inference(all_rule,[status(thm)],]) ).
thf(15,plain,
( sP23
| ~ sP27 ),
inference(prop_rule,[status(thm)],]) ).
thf(16,plain,
( sP23
| sP29 ),
inference(prop_rule,[status(thm)],]) ).
thf(17,plain,
( ~ sP15
| ~ sP23
| sP21 ),
inference(prop_rule,[status(thm)],]) ).
thf(18,plain,
( ~ sP17
| sP15 ),
inference(all_rule,[status(thm)],]) ).
thf(19,plain,
( ~ sP18
| sP7 ),
inference(all_rule,[status(thm)],]) ).
thf(20,plain,
( ~ sP18
| sP4 ),
inference(all_rule,[status(thm)],]) ).
thf(21,plain,
( ~ sP25
| sP10 ),
inference(all_rule,[status(thm)],]) ).
thf(22,plain,
( ~ sP11
| sP17 ),
inference(all_rule,[status(thm)],]) ).
thf(23,plain,
( ~ sP6
| sP18 ),
inference(all_rule,[status(thm)],]) ).
thf(24,plain,
( ~ sP28
| sP25 ),
inference(all_rule,[status(thm)],]) ).
thf(25,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,h1,h3,h5,h7,h9,h10]) ).
thf(26,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h7,h8,h5,h6,h3,h4,h1,h2,h0,h0]),tab_negimp(discharge,[h9,h10])],[h8,25,h9,h10]) ).
thf(27,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h3,h4,h1,h2,h0,h0]),tab_negimp(discharge,[h7,h8])],[h6,26,h7,h8]) ).
thf(28,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h4,h1,h2,h0,h0]),tab_negimp(discharge,[h5,h6])],[h4,27,h5,h6]) ).
thf(29,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h2,h0,h0]),tab_negimp(discharge,[h3,h4])],[h2,28,h3,h4]) ).
thf(30,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h0,h0]),tab_negimp(discharge,[h1,h2])],[h0,29,h1,h2]) ).
thf(0,theorem,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( sP6
=> ( sP28
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[30,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : SEU711^2 : TPTP v8.1.2. Released v3.7.0.
% 0.00/0.10 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.11/0.29 % Computer : n032.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Wed Aug 23 12:35:40 EDT 2023
% 0.11/0.30 % CPUTime :
% 92.52/92.80 % SZS status Theorem
% 92.52/92.80 % Mode: cade22grackle2x4fb9
% 92.52/92.80 % Steps: 108739
% 92.52/92.80 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------