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Mirrors > Home > MPE Home > Th. List > encv | Structured version Visualization version 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 8927 | . 2 ⊢ Rel ≈ | |
2 | 1 | brrelex12i 5723 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 Vcvv 3473 class class class wbr 5141 ≈ cen 8919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-en 8923 |
This theorem is referenced by: bren 8932 brenOLD 8933 en0 8996 en0r 8999 en1 9004 rexdif1en 9141 dif1en 9143 enp1i 9262 ensucne0OLD 42052 axccd 43699 |
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