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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 6109 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) | |
2 | 1 | ancoms 458 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
3 | df-ec 8746 | . . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 3 | eleq2i 2831 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵})) |
5 | df-br 5149 | . 2 ⊢ (𝐵𝑅𝐴 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | |
6 | 2, 4, 5 | 3bitr4g 314 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 {csn 4631 〈cop 4637 class class class wbr 5148 “ cima 5692 [cec 8742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 |
This theorem is referenced by: ecref 8789 elec 8790 relelec 8791 ecdmn0 8793 erth 8795 erdisj 8798 qsel 8835 ghmqusnsglem1 19311 ghmquskerlem1 19314 orbsta 19344 sylow2alem1 19650 sylow2blem1 19653 sylow3lem3 19662 efgi2 19758 rngqiprngfulem2 21340 rngqipring1 21344 tgpconncompeqg 24136 xmetec 24460 blpnfctr 24462 xmetresbl 24463 xrsblre 24847 ecxpid 33369 lsmsnorb 33399 ecin0 38334 eqvrelth 38593 qsalrel 42260 |
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