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Mirrors > Home > ILE Home > Th. List > ecid | GIF version |
Description: A set is equal to its converse epsilon coset. (Note: converse epsilon is not an equivalence relation.) (Contributed by NM, 13-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ecid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ecid | ⊢ [𝐴]◡ E = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | ecid.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | elec 6468 | . . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦) |
4 | 2, 1 | brcnv 4722 | . . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴) |
5 | 2 | epelc 4213 | . . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
6 | 3, 4, 5 | 3bitri 205 | . 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴) |
7 | 6 | eqriv 2136 | 1 ⊢ [𝐴]◡ E = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2686 class class class wbr 3929 E cep 4209 ◡ccnv 4538 [cec 6427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-eprel 4211 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-ec 6431 |
This theorem is referenced by: qsid 6494 |
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