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Theorem eqeng 7133
Description: Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
eqeng  |-  ( A  e.  V  ->  ( A  =  B  ->  A 
~~  B ) )

Proof of Theorem eqeng
StepHypRef Expression
1 enrefg 7131 . 2  |-  ( A  e.  V  ->  A  ~~  A )
2 breq2 4208 . 2  |-  ( A  =  B  ->  ( A  ~~  A  <->  A  ~~  B ) )
31, 2syl5ibcom 212 1  |-  ( A  e.  V  ->  ( A  =  B  ->  A 
~~  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   class class class wbr 4204    ~~ cen 7098
This theorem is referenced by:  idssen  7144  nneneq  7282  onomeneq  7288  pr2ne  7881  alephord  7948  alephdom  7954  fin23lem25  8196  alephadd  8444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-en 7102
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