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Theorem fisbth 6829
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 6707 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32ad2antrr 480 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6707 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 119 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad3antlr 485 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrr 526 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  ~~  n )
87ensymd 6729 . . . . . . . . 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 521 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~<_  B )
109ad2antrr 480 . . . . . . . . 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 6736 . . . . . . . . 9  |-  ( ( n  ~~  A  /\  A  ~<_  B )  ->  n  ~<_  B )
128, 10, 11syl2anc 409 . . . . . . . 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 522 . . . . . . . 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 6737 . . . . . . . 8  |-  ( ( n  ~<_  B  /\  B  ~~  m )  ->  n  ~<_  m )
1512, 13, 14syl2anc 409 . . . . . . 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 525 . . . . . . . 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 521 . . . . . . . 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 6810 . . . . . . . 8  |-  ( ( n  e.  om  /\  m  e.  om )  ->  ( n  ~<_  m  <->  n  C_  m
) )
1916, 17, 18syl2anc 409 . . . . . . 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 6729 . . . . . . . . 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 522 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  B  ~<_  A )
2322ad2antrr 480 . . . . . . . . 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 6736 . . . . . . . . 9  |-  ( ( m  ~~  B  /\  B  ~<_  A )  ->  m  ~<_  A )
2521, 23, 24syl2anc 409 . . . . . . . 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 6737 . . . . . . . 8  |-  ( ( m  ~<_  A  /\  A  ~~  n )  ->  m  ~<_  n )
2725, 7, 26syl2anc 409 . . . . . . 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 6810 . . . . . . . 8  |-  ( ( m  e.  om  /\  n  e.  om )  ->  ( m  ~<_  n  <->  m  C_  n
) )
2917, 16, 28syl2anc 409 . . . . . . 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 3145 . . . . 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 3991 . . . 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 6730 . . . 4  |-  ( ( A  ~~  m  /\  m  ~~  B )  ->  A  ~~  B )
3432, 21, 33syl2anc 409 . . 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 2579 . 2  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  A  ~~  B )
363, 35rexlimddv 2579 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 2128   E.wrex 2436    C_ wss 3102   class class class wbr 3966   omcom 4550    ~~ cen 6684    ~<_ cdom 6685   Fincfn 6686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-br 3967  df-opab 4027  df-tr 4064  df-id 4254  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-er 6481  df-en 6687  df-dom 6688  df-fin 6689
This theorem is referenced by: (None)
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