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Theorem fisbth 6861
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 6739 . . . 4  |-  ( A  e.  Fin  <->  E. n  e.  om  A  ~~  n
)
21biimpi 119 . . 3  |-  ( A  e.  Fin  ->  E. n  e.  om  A  ~~  n
)
32ad2antrr 485 . 2  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  E. n  e.  om  A  ~~  n )
4 isfi 6739 . . . . 5  |-  ( B  e.  Fin  <->  E. m  e.  om  B  ~~  m
)
54biimpi 119 . . . 4  |-  ( B  e.  Fin  ->  E. m  e.  om  B  ~~  m
)
65ad3antlr 490 . . 3  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  E. m  e.  om  B  ~~  m
)
7 simplrr 531 . . . . 5  |-  ( ( ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  /\  ( m  e.  om  /\  B  ~~  m ) )  ->  A  ~~  n )
87ensymd 6761 . . . . . . . . 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 526 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  A  ~<_  B )
109ad2antrr 485 . . . . . . . . 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 6768 . . . . . . . . 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 527 . . . . . . . 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 6769 . . . . . . . 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 530 . . . . . . . 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 526 . . . . . . . 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 6842 . . . . . . . 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 6761 . . . . . . . . 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 527 . . . . . . . . . 10  |-  ( ( ( A  e.  Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  ->  B  ~<_  A )
2322ad2antrr 485 . . . . . . . . 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 6768 . . . . . . . . 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 6769 . . . . . . . 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 6842 . . . . . . . 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 3164 . . . . 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 4015 . . . 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 6762 . . . 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 2592 . 2  |-  ( ( ( ( A  e. 
Fin  /\  B  e.  Fin )  /\  ( A  ~<_  B  /\  B  ~<_  A ) )  /\  ( n  e.  om  /\  A  ~~  n ) )  ->  A  ~~  B )
363, 35rexlimddv 2592 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 2141   E.wrex 2449    C_ wss 3121   class class class wbr 3989   omcom 4574    ~~ cen 6716    ~<_ cdom 6717   Fincfn 6718
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721
This theorem is referenced by: (None)
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