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Theorem encv 8503
 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 8500 . 2 Rel ≈
21brrelex12i 5572 1 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3441   class class class wbr 5031   ≈ cen 8492 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-en 8496 This theorem is referenced by:  bren  8504  ensucne0OLD  40281  axccd  41904
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