ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  encv GIF version

Theorem encv 6570
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 6568 . 2 Rel ≈
2 brrelex12 4515 . 2 ((Rel ≈ ∧ 𝐴𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2mpan 418 1 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1448  Vcvv 2641   class class class wbr 3875  Rel wrel 4482  cen 6562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-en 6565
This theorem is referenced by:  bren  6571  en1uniel  6628  cardcl  6948  isnumi  6949  cardval3ex  6952  djuen  6971
  Copyright terms: Public domain W3C validator