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Theorem domen2 6489
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 6438 . . 3  |-  ( ( C  ~<_  A  /\  A  ~~  B )  ->  C  ~<_  B )
21ancoms 264 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  A )  ->  C  ~<_  B )
3 ensym 6428 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 domentr 6438 . . . 4  |-  ( ( C  ~<_  B  /\  B  ~~  A )  ->  C  ~<_  A )
54ancoms 264 . . 3  |-  ( ( B  ~~  A  /\  C  ~<_  B )  ->  C  ~<_  A )
63, 5sylan 277 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  B )  ->  C  ~<_  A )
72, 6impbida 561 1  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   class class class wbr 3811    ~~ cen 6385    ~<_ cdom 6386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-er 6222  df-en 6388  df-dom 6389
This theorem is referenced by:  fihashdom  10046
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