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Theorem domen2 7250
Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
Assertion
Ref Expression
domen2  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )

Proof of Theorem domen2
StepHypRef Expression
1 domentr 7166 . . 3  |-  ( ( C  ~<_  A  /\  A  ~~  B )  ->  C  ~<_  B )
21ancoms 440 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  A )  ->  C  ~<_  B )
3 ensym 7156 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 domentr 7166 . . . 4  |-  ( ( C  ~<_  B  /\  B  ~~  A )  ->  C  ~<_  A )
54ancoms 440 . . 3  |-  ( ( B  ~~  A  /\  C  ~<_  B )  ->  C  ~<_  A )
63, 5sylan 458 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  B )  ->  C  ~<_  A )
72, 6impbida 806 1  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   class class class wbr 4212    ~~ cen 7106    ~<_ cdom 7107
This theorem is referenced by:  fisucdomOLD  7312  infdiffi  7612  carddomi2  7857  numdom  7919  cdadom2  8067  infdif  8089  fin45  8272  fin67  8275  aleph1  8446  gchdomtri  8504  gchhar  8546  gchpwdom  8549  ctbnfien  26879
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-er 6905  df-en 7110  df-dom 7111
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