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Mirrors > Home > ILE Home > Th. List > encv | GIF version |
Description: If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.) |
Ref | Expression |
---|---|
encv | ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 6737 | . 2 ⊢ Rel ≈ | |
2 | brrelex12 4660 | . 2 ⊢ ((Rel ≈ ∧ 𝐴 ≈ 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | 1, 2 | mpan 424 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 Vcvv 2737 class class class wbr 4000 Rel wrel 4627 ≈ cen 6731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4628 df-rel 4629 df-en 6734 |
This theorem is referenced by: bren 6740 en1uniel 6797 cardcl 7173 isnumi 7174 cardval3ex 7177 djuen 7203 ccfunen 7241 |
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