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Mirrors > Home > MPE Home > Th. List > epeli | Structured version Visualization version GIF version |
Description: The membership relation and the membership predicate agree when the "containing" class is a set. Inference associated with epelg 5580. (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 5580 | . 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 205 ∈ wcel 2107 Vcvv 3475 class class class wbr 5147 E cep 5578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-eprel 5579 |
This theorem is referenced by: epel 5582 0sn0ep 5583 smoiso 8357 smoiso2 8364 ecid 8772 ordiso2 9506 cantnflt 9663 cantnfp1lem3 9671 oemapso 9673 cantnflem1b 9677 cantnflem1 9680 cantnf 9684 wemapwe 9688 cnfcomlem 9690 cnfcom 9691 cnfcom3lem 9694 leweon 10002 r0weon 10003 alephiso 10089 fin23lem27 10319 fpwwe2lem8 10629 ex-eprel 29666 cardpred 34031 satefvfmla0 34347 satefvfmla1 34354 dftr6 34659 coep 34660 coepr 34661 brsset 34799 brtxpsd 34804 brcart 34842 dfrecs2 34860 dfrdg4 34861 cnambfre 36474 wepwsolem 41717 dnwech 41723 |
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