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Theorem sdomen1 7188
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 7093 . . 3  |-  ( A 
~~  B  ->  B  ~~  A )
2 ensdomtr 7180 . . 3  |-  ( ( B  ~~  A  /\  A  ~<  C )  ->  B  ~<  C )
31, 2sylan 458 . 2  |-  ( ( A  ~~  B  /\  A  ~<  C )  ->  B  ~<  C )
4 ensdomtr 7180 . 2  |-  ( ( A  ~~  B  /\  B  ~<  C )  ->  A  ~<  C )
53, 4impbida 806 1  |-  ( A 
~~  B  ->  ( A  ~<  C  <->  B  ~<  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   class class class wbr 4154    ~~ cen 7043    ~< csdm 7045
This theorem is referenced by:  isfiniteg  7304  cdafi  8004  alephval2  8381  engch  8437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049
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