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Theorem sdomen1 7454
Description: Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
Assertion
Ref Expression
sdomen1  |-  ( A 
~~  B  ->  ( A  ~<  C  <->  B  ~<  C ) )

Proof of Theorem sdomen1
StepHypRef Expression
1 ensym 7357 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
2 ensdomtr 7446 . . 3  |-  ( ( B  ~~  A  /\  A  ~<  C )  ->  B  ~<  C )
31, 2sylan 471 . 2  |-  ( ( A  ~~  B  /\  A  ~<  C )  ->  B  ~<  C )
4 ensdomtr 7446 . 2  |-  ( ( A  ~~  B  /\  B  ~<  C )  ->  A  ~<  C )
53, 4impbida 828 1  |-  ( A 
~~  B  ->  ( A  ~<  C  <->  B  ~<  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4291    ~~ cen 7306    ~< csdm 7308
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 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312
This theorem is referenced by:  isfiniteg  7571  cdafi  8358  alephval2  8735  engch  8794
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