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Theorem sdomen1 7653
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 7556 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
2 ensdomtr 7645 . . 3  |-  ( ( B  ~~  A  /\  A  ~<  C )  ->  B  ~<  C )
31, 2sylan 471 . 2  |-  ( ( A  ~~  B  /\  A  ~<  C )  ->  B  ~<  C )
4 ensdomtr 7645 . 2  |-  ( ( A  ~~  B  /\  B  ~<  C )  ->  A  ~<  C )
53, 4impbida 829 1  |-  ( A 
~~  B  ->  ( A  ~<  C  <->  B  ~<  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   class class class wbr 4442    ~~ cen 7505    ~< csdm 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511
This theorem is referenced by:  isfiniteg  7771  cdafi  8561  alephval2  8938  engch  8997
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