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Mirrors > Home > ILE Home > Th. List > epel | GIF version |
Description: The epsilon relation and the membership relation are the same. (Contributed by NM, 13-Aug-1995.) |
Ref | Expression |
---|---|
epel | ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2615 | . 2 ⊢ 𝑦 ∈ V | |
2 | 1 | epelc 4081 | 1 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 class class class wbr 3811 E cep 4077 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-eprel 4079 |
This theorem is referenced by: epse 4132 wetrep 4150 ordsoexmid 4340 zfregfr 4351 ordwe 4353 wessep 4355 reg3exmidlemwe 4356 smoiso 5997 nnwetri 6551 ordiso2 6633 frec2uzisod 9701 |
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