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Theorem encv 6647
 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 6645 . 2 Rel ≈
2 brrelex12 4584 . 2 ((Rel ≈ ∧ 𝐴𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
31, 2mpan 421 1 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1481  Vcvv 2689   class class class wbr 3936  Rel wrel 4551   ≈ cen 6639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-rel 4553  df-en 6642 This theorem is referenced by:  bren  6648  en1uniel  6705  cardcl  7053  isnumi  7054  cardval3ex  7057  djuen  7083  ccfunen  7095
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