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Theorem eqeng 8257
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 8255 . 2 (𝐴𝑉𝐴𝐴)
2 breq2 4878 . 2 (𝐴 = 𝐵 → (𝐴𝐴𝐴𝐵))
31, 2syl5ibcom 237 1 (𝐴𝑉 → (𝐴 = 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166   class class class wbr 4874  cen 8220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-en 8224
This theorem is referenced by:  idssen  8268  nneneq  8413  onomeneq  8420  pr2ne  9142  alephord  9212  alephdom  9218  fin23lem25  9462  alephadd  9715  rp-isfinite5  38705
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