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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 8990 | . 2 ⊢ Rel ≈ | |
| 2 | 1 | brrelex12i 5740 | 1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ≈ cen 8982 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-en 8986 | 
| This theorem is referenced by: bren 8995 en0 9058 en0r 9060 en1 9064 rexdif1en 9198 dif1en 9200 enp1i 9313 ensucne0OLD 43543 axccd 45234 | 
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