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Theorem domen2 7618
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 7532 . . 3  |-  ( ( C  ~<_  A  /\  A  ~~  B )  ->  C  ~<_  B )
21ancoms 451 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  A )  ->  C  ~<_  B )
3 ensym 7522 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 domentr 7532 . . . 4  |-  ( ( C  ~<_  B  /\  B  ~~  A )  ->  C  ~<_  A )
54ancoms 451 . . 3  |-  ( ( B  ~~  A  /\  C  ~<_  B )  ->  C  ~<_  A )
63, 5sylan 469 . 2  |-  ( ( A  ~~  B  /\  C  ~<_  B )  ->  C  ~<_  A )
72, 6impbida 833 1  |-  ( A 
~~  B  ->  ( C  ~<_  A  <->  C  ~<_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4394    ~~ cen 7471    ~<_ cdom 7472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-er 7268  df-en 7475  df-dom 7476
This theorem is referenced by:  infdiffi  8027  carddomi2  8303  numdom  8371  cdadom2  8519  infdif  8541  fin45  8724  fin67  8727  aleph1  8898  gchdomtri  8957  gchpwdom  8998  gchhar  9007  ctbnfien  35094
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