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Theorem sdomen2 7456
Description: Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
Assertion
Ref Expression
sdomen2  |-  ( A 
~~  B  ->  ( C  ~<  A  <->  C  ~<  B ) )

Proof of Theorem sdomen2
StepHypRef Expression
1 sdomentr 7445 . . 3  |-  ( ( C  ~<  A  /\  A  ~~  B )  ->  C  ~<  B )
21ancoms 453 . 2  |-  ( ( A  ~~  B  /\  C  ~<  A )  ->  C  ~<  B )
3 ensym 7358 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
4 sdomentr 7445 . . . 4  |-  ( ( C  ~<  B  /\  B  ~~  A )  ->  C  ~<  A )
54ancoms 453 . . 3  |-  ( ( B  ~~  A  /\  C  ~<  B )  ->  C  ~<  A )
63, 5sylan 471 . 2  |-  ( ( A  ~~  B  /\  C  ~<  B )  ->  C  ~<  A )
72, 6impbida 828 1  |-  ( A 
~~  B  ->  ( C  ~<  A  <->  C  ~<  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4292    ~~ cen 7307    ~< csdm 7309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313
This theorem is referenced by:  fisucdomOLD  7516  cdaxpdom  8358  alephval2  8736  engch  8795  canthp1lem2  8820  hargch  8840  alephgch  8841  ovoliunnfl  28433
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