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Mirrors > Home > ILE Home > Th. List > relelec | GIF version |
Description: Membership in an equivalence class when 𝑅 is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
relelec | ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2631 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐴 ∈ V) | |
2 | ecexr 6311 | . . . 4 ⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) | |
3 | 1, 2 | jca 301 | . . 3 ⊢ (𝐴 ∈ [𝐵]𝑅 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
4 | 3 | adantl 272 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ [𝐵]𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
5 | brrelex12 4489 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
6 | 5 | ancomd 264 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐴) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
7 | elecg 6344 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) | |
8 | 4, 6, 7 | pm5.21nd 864 | 1 ⊢ (Rel 𝑅 → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1439 Vcvv 2620 class class class wbr 3851 Rel wrel 4457 [cec 6304 |
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 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-sbc 2842 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-xp 4458 df-rel 4459 df-cnv 4460 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-ec 6308 |
This theorem is referenced by: (None) |
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