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Theorem fisbth 6653
Description: Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.)
Assertion
Ref Expression
fisbth  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~~  B
)

Proof of Theorem fisbth
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6532 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32ad2antrr 473 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6532 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 119 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad3antlr 478 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrr 504 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  ~~  n )
87ensymd 6554 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  ~~  A )
9 simprl 499 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~<_  B )
109ad2antrr 473 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  ~<_  B )
11 endomtr 6561 . . . . . . . . 9  |-  ( ( n  ~~  A  /\  A  ~<_  B )  ->  n  ~<_  B )
128, 10, 11syl2anc 404 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  ~<_  B )
13 simprr 500 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  B  ~~  m )
14 domentr 6562 . . . . . . . 8  |-  ( ( n  ~<_  B  /\  B  ~~  m )  ->  n  ~<_  m )
1512, 13, 14syl2anc 404 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  ~<_  m )
16 simplrl 503 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  e.  om )
17 simprl 499 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  m  e.  om )
18 nndomo 6634 . . . . . . . 8  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  ~<_  m  <->  n  C_  m
) )
1916, 17, 18syl2anc 404 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( n  ~<_  m  <->  n  C_  m
) )
2015, 19mpbid 146 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  C_  m )
2113ensymd 6554 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  m  ~~  B )
22 simprr 500 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  B  ~<_  A )
2322ad2antrr 473 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  B  ~<_  A )
24 endomtr 6561 . . . . . . . . 9  |-  ( ( m  ~~  B  /\  B  ~<_  A )  ->  m  ~<_  A )
2521, 23, 24syl2anc 404 . . . . . . . 8  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  m  ~<_  A )
26 domentr 6562 . . . . . . . 8  |-  ( ( m  ~<_  A  /\  A  ~~  n )  ->  m  ~<_  n )
2725, 7, 26syl2anc 404 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  m  ~<_  n )
28 nndomo 6634 . . . . . . . 8  |-  ( ( m  e.  om  /\  n  e.  om )  ->  ( m  ~<_  n  <->  m  C_  n
) )
2917, 16, 28syl2anc 404 . . . . . . 7  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  -> 
( m  ~<_  n  <->  m  C_  n
) )
3027, 29mpbid 146 . . . . . 6  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  m  C_  n )
3120, 30eqssd 3043 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  n  =  m )
327, 31breqtrd 3875 . . . 4  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  ~~  m )
33 entr 6555 . . . 4  |-  ( ( A  ~~  m  /\  m  ~~  B )  ->  A  ~~  B )
3432, 21, 33syl2anc 404 . . 3  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  ~~  B )
356, 34rexlimddv 2494 . 2  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  A  ~~  B )
363, 35rexlimddv 2494 1  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~~  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1439   E.wrex 2361    C_ wss 3000   class class class wbr 3851   omcom 4418    ~~ cen 6509    ~<_ cdom 6510   Fincfn 6511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514
This theorem is referenced by: (None)
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