Theorem encv 8953

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 8950	. 2 ⊢ Rel ≈
2	1	brrelex12i 5731	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105  Vcvv 3473   class class class wbr 5148   ≈ cen 8942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-en 8946

This theorem is referenced by:  bren  8955  brenOLD  8956  en0  9019  en0r  9022  en1  9027  rexdif1en  9164  dif1en  9166  enp1i  9285  ensucne0OLD  42596  axccd  44239