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Theorem sbthlem10 7422
Description: Lemma for sbth 7423. (Contributed by NM, 28-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1  |-  A  e. 
_V
sbthlem.2  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
sbthlem.3  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
sbthlem.4  |-  B  e. 
_V
Assertion
Ref Expression
sbthlem10  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
Distinct variable groups:    x, A    x, B    x, D    x, f, g    x, H    f,
g, A    B, f,
g
Allowed substitution hints:    D( f, g)    H( f, g)

Proof of Theorem sbthlem10
StepHypRef Expression
1 sbthlem.4 . . . . 5  |-  B  e. 
_V
21brdom 7314 . . . 4  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 sbthlem.1 . . . . 5  |-  A  e. 
_V
43brdom 7314 . . . 4  |-  ( B  ~<_  A  <->  E. g  g : B -1-1-> A )
52, 4anbi12i 697 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  <->  ( E. f  f : A -1-1-> B  /\  E. g  g : B -1-1-> A ) )
6 eeanv 1931 . . 3  |-  ( E. f E. g ( f : A -1-1-> B  /\  g : B -1-1-> A
)  <->  ( E. f 
f : A -1-1-> B  /\  E. g  g : B -1-1-> A ) )
75, 6bitr4i 252 . 2  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  <->  E. f E. g ( f : A -1-1-> B  /\  g : B -1-1-> A ) )
8 sbthlem.3 . . . . 5  |-  H  =  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A 
\  U. D ) ) )
9 vex 2970 . . . . . . 7  |-  f  e. 
_V
109resex 5145 . . . . . 6  |-  ( f  |`  U. D )  e. 
_V
11 vex 2970 . . . . . . . 8  |-  g  e. 
_V
1211cnvex 6520 . . . . . . 7  |-  `' g  e.  _V
1312resex 5145 . . . . . 6  |-  ( `' g  |`  ( A  \ 
U. D ) )  e.  _V
1410, 13unex 6373 . . . . 5  |-  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D ) ) )  e.  _V
158, 14eqeltri 2508 . . . 4  |-  H  e. 
_V
16 sbthlem.2 . . . . 5  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
173, 16, 8sbthlem9 7421 . . . 4  |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H : A
-1-1-onto-> B )
18 f1oen3g 7317 . . . 4  |-  ( ( H  e.  _V  /\  H : A -1-1-onto-> B )  ->  A  ~~  B )
1915, 17, 18sylancr 663 . . 3  |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  A  ~~  B )
2019exlimivv 1689 . 2  |-  ( E. f E. g ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  A  ~~  B )
217, 20sylbi 195 1  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2424   _Vcvv 2967    \ cdif 3320    u. cun 3321    C_ wss 3323   U.cuni 4086   class class class wbr 4287   `'ccnv 4834    |` cres 4837   "cima 4838   -1-1->wf1 5410   -1-1-onto->wf1o 5412    ~~ cen 7299    ~<_ cdom 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-en 7303  df-dom 7304
This theorem is referenced by:  sbth  7423
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