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Mirrors > Home > MPE Home > Th. List > epelc | Structured version Visualization version GIF version |
Description: The epsilon relationship and the membership relation are the same. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
epelc.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
epelc | ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epelc.1 | . 2 ⊢ 𝐵 ∈ V | |
2 | epelg 5030 | . 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 196 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 E cep 5028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-eprel 5029 |
This theorem is referenced by: epel 5032 epini 5495 smoiso 7459 smoiso2 7466 ecid 7812 ordiso2 8420 oismo 8445 cantnflt 8569 cantnfp1lem3 8577 oemapso 8579 cantnflem1b 8583 cantnflem1 8586 cantnf 8590 wemapwe 8594 cnfcomlem 8596 cnfcom 8597 cnfcom3lem 8600 leweon 8834 r0weon 8835 alephiso 8921 fin23lem27 9150 fpwwe2lem9 9460 ex-eprel 27290 dftr6 31640 coep 31641 coepr 31642 brsset 31996 brtxpsd 32001 brcart 32039 dfrecs2 32057 dfrdg4 32058 cnambfre 33458 wepwsolem 37612 dnwech 37618 |
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