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| Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5585. (Contributed by Scott Fenton, 11-Apr-2012.) | 
| Ref | Expression | 
|---|---|
| epeli.1 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| epeli | ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | epeli.1 | . 2 ⊢ 𝐵 ∈ V | |
| 2 | epelg 5585 | . 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 E cep 5583 | 
| 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-ne 2941 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-eprel 5584 | 
| This theorem is referenced by: epel 5587 0sn0ep 5588 smoiso 8402 smoiso2 8409 ecid 8822 ordiso2 9555 cantnflt 9712 cantnfp1lem3 9720 oemapso 9722 cantnflem1b 9726 cantnflem1 9729 cantnf 9733 wemapwe 9737 cnfcomlem 9739 cnfcom 9740 cnfcom3lem 9743 leweon 10051 r0weon 10052 alephiso 10138 fin23lem27 10368 fpwwe2lem8 10678 ex-eprel 30452 cardpred 35104 satefvfmla0 35423 satefvfmla1 35430 dftr6 35751 coep 35752 coepr 35753 brsset 35890 brtxpsd 35895 brcart 35933 dfrecs2 35951 dfrdg4 35952 cnambfre 37675 wepwsolem 43054 dnwech 43060 rankrelp 44977 | 
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