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Theorem encv 8887
Description: If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
Assertion
Ref Expression
encv (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem encv
StepHypRef Expression
1 relen 8884 . 2 Rel ≈
21brrelex12i 5685 1 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3443   class class class wbr 5103  cen 8876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-en 8880
This theorem is referenced by:  bren  8889  brenOLD  8890  en0  8953  en0r  8956  en1  8961  rexdif1en  9098  dif1en  9100  enp1i  9219  ensucne0OLD  41744  axccd  43386
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