Theorem encv 8992

Description: If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)

Assertion

Ref	Expression
encv	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem encv

Step	Hyp	Ref	Expression
1		relen 8989	. 2 ⊢ Rel ≈
2	1	brrelex12i 5744	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148   ≈ cen 8981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-en 8985

This theorem is referenced by:  bren  8994  en0  9057  en0r  9059  en1  9063  rexdif1en  9197  dif1en  9199  enp1i  9311  ensucne0OLD  43520  axccd  45172