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Mirrors > Home > ILE Home > Th. List > epelc | 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 4170 | . 2 ⊢ (𝐵 ∈ V → (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 E 𝐵 ↔ 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1461 Vcvv 2655 class class class wbr 3893 E cep 4167 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 df-opab 3948 df-eprel 4169 |
This theorem is referenced by: epel 4172 epini 4866 ecid 6444 ordiso2 6870 |
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