Theorem encv 8880

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 8877	. 2 ⊢ Rel ≈
2	1	brrelex12i 5674	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3436   class class class wbr 5092   ≈ cen 8869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-en 8873

This theorem is referenced by:  bren  8882  en0  8943  en0r  8945  en1  8949  rexdif1en  9074  dif1en  9075  enp1i  9168  ensucne0OLD  43503  axccd  45207