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Theorem eqeng 7848
Description: Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
Assertion
Ref Expression
eqeng (𝐴𝑉 → (𝐴 = 𝐵𝐴𝐵))

Proof of Theorem eqeng
StepHypRef Expression
1 enrefg 7846 . 2 (𝐴𝑉𝐴𝐴)
2 breq2 4577 . 2 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2syl5ibcom 233 1 (𝐴𝑉 → (𝐴 = 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975   class class class wbr 4573  cen 7811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-en 7815
This theorem is referenced by:  idssen  7859  nneneq  8001  onomeneq  8008  pr2ne  8684  alephord  8754  alephdom  8760  fin23lem25  9002  alephadd  9251  rp-isfinite5  36681
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