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Theorem sbthlem10 7155
Description: Lemma for sbth 7156. (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 7049 . . . 4  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 sbthlem.1 . . . . 5  |-  A  e. 
_V
43brdom 7049 . . . 4  |-  ( B  ~<_  A  <->  E. g  g : B -1-1-> A )
52, 4anbi12i 679 . . 3  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  <->  ( E. f  f : A -1-1-> B  /\  E. g  g : B -1-1-> A ) )
6 eeanv 1926 . . 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 244 . 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 2895 . . . . . . 7  |-  f  e. 
_V
109resex 5119 . . . . . 6  |-  ( f  |`  U. D )  e. 
_V
11 vex 2895 . . . . . . . 8  |-  g  e. 
_V
1211cnvex 5339 . . . . . . 7  |-  `' g  e.  _V
1312resex 5119 . . . . . 6  |-  ( `' g  |`  ( A  \ 
U. D ) )  e.  _V
1410, 13unex 4640 . . . . 5  |-  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D ) ) )  e.  _V
158, 14eqeltri 2450 . . . 4  |-  H  e. 
_V
16 sbthlem.2 . . . . 5  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
173, 16, 8sbthlem9 7154 . . . 4  |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H : A
-1-1-onto-> B )
18 f1oen3g 7052 . . . 4  |-  ( ( H  e.  _V  /\  H : A -1-1-onto-> B )  ->  A  ~~  B )
1915, 17, 18sylancr 645 . . 3  |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  A  ~~  B )
2019exlimivv 1642 . 2  |-  ( E. f E. g ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  A  ~~  B )
217, 20sylbi 188 1  |-  ( ( A  ~<_  B  /\  B  ~<_  A )  ->  A  ~~  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2366   _Vcvv 2892    \ cdif 3253    u. cun 3254    C_ wss 3256   U.cuni 3950   class class class wbr 4146   `'ccnv 4810    |` cres 4813   "cima 4814   -1-1->wf1 5384   -1-1-onto->wf1o 5386    ~~ cen 7035    ~<_ cdom 7036
This theorem is referenced by:  sbth  7156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-en 7039  df-dom 7040
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