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Mirrors > Home > MPE Home > Th. List > ecid | Structured version Visualization version 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 3401 | . . . 4 ⊢ 𝑦 ∈ V | |
2 | ecid.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | 1, 2 | elec 8068 | . . 3 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝐴◡ E 𝑦) |
4 | 2, 1 | brcnv 5550 | . . 3 ⊢ (𝐴◡ E 𝑦 ↔ 𝑦 E 𝐴) |
5 | 2 | epeli 5268 | . . 3 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
6 | 3, 4, 5 | 3bitri 289 | . 2 ⊢ (𝑦 ∈ [𝐴]◡ E ↔ 𝑦 ∈ 𝐴) |
7 | 6 | eqriv 2775 | 1 ⊢ [𝐴]◡ E = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 Vcvv 3398 class class class wbr 4886 E cep 5265 ◡ccnv 5354 [cec 8024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-eprel 5266 df-xp 5361 df-cnv 5363 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ec 8028 |
This theorem is referenced by: qsid 8096 addcnsrec 10300 mulcnsrec 10301 |
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