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Theorem sbthlem10 7218
Description: Lemma for sbth 7219. (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 7112 . . . 4  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
3 sbthlem.1 . . . . 5  |-  A  e. 
_V
43brdom 7112 . . . 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 1937 . . 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 2951 . . . . . . 7  |-  f  e. 
_V
109resex 5178 . . . . . 6  |-  ( f  |`  U. D )  e. 
_V
11 vex 2951 . . . . . . . 8  |-  g  e. 
_V
1211cnvex 5398 . . . . . . 7  |-  `' g  e.  _V
1312resex 5178 . . . . . 6  |-  ( `' g  |`  ( A  \ 
U. D ) )  e.  _V
1410, 13unex 4699 . . . . 5  |-  ( ( f  |`  U. D )  u.  ( `' g  |`  ( A  \  U. D ) ) )  e.  _V
158, 14eqeltri 2505 . . . 4  |-  H  e. 
_V
16 sbthlem.2 . . . . 5  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
173, 16, 8sbthlem9 7217 . . . 4  |-  ( ( f : A -1-1-> B  /\  g : B -1-1-> A
)  ->  H : A
-1-1-onto-> B )
18 f1oen3g 7115 . . . 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 1645 . 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   U.cuni 4007   class class class wbr 4204   `'ccnv 4869    |` cres 4872   "cima 4873   -1-1->wf1 5443   -1-1-onto->wf1o 5445    ~~ cen 7098    ~<_ cdom 7099
This theorem is referenced by:  sbth  7219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-en 7102  df-dom 7103
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