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Mirrors > Home > MPE Home > Th. List > elecg | Structured version Visualization version GIF version |
Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
elecg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimasng 5940 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) | |
2 | 1 | ancoms 462 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
3 | df-ec 8371 | . . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 3 | eleq2i 2822 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵})) |
5 | df-br 5040 | . 2 ⊢ (𝐵𝑅𝐴 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | |
6 | 2, 4, 5 | 3bitr4g 317 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2112 {csn 4527 〈cop 4533 class class class wbr 5039 “ cima 5539 [cec 8367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ec 8371 |
This theorem is referenced by: elec 8413 relelec 8414 ecdmn0 8416 erth 8418 erdisj 8421 qsel 8456 orbsta 18661 sylow2alem1 18960 sylow2blem1 18963 sylow3lem3 18972 efgi2 19069 tgpconncompeqg 22963 xmetec 23286 blpnfctr 23288 xmetresbl 23289 xrsblre 23662 ecxpid 31224 lsmsnorb 31247 ecin0 36170 eqvrelth 36410 qsalrel 39869 |
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