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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 2560 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | ecid.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | elec 6145 | . . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦) |
4 | 2, 1 | brcnv 4518 | . . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴) |
5 | 2 | epelc 4028 | . . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
6 | 3, 4, 5 | 3bitri 195 | . 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴) |
7 | 6 | eqriv 2037 | 1 ⊢ [𝐴]◡ E = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 Vcvv 2557 class class class wbr 3764 E cep 4024 ◡ccnv 4344 [cec 6104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-eprel 4026 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-ec 6108 |
This theorem is referenced by: qsid 6171 |
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