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| 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 6106 | . . 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 8748 | . . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
| 4 | 3 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵})) | 
| 5 | df-br 5143 | . 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 2107 {csn 4625 〈cop 4631 class class class wbr 5142 “ cima 5687 [cec 8744 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ec 8748 | 
| This theorem is referenced by: ecref 8791 elec 8792 relelec 8793 ecdmn0 8795 erth 8797 erdisj 8800 qsel 8837 ghmqusnsglem1 19299 ghmquskerlem1 19302 orbsta 19332 sylow2alem1 19636 sylow2blem1 19639 sylow3lem3 19648 efgi2 19744 rngqiprngfulem2 21323 rngqipring1 21327 tgpconncompeqg 24121 xmetec 24445 blpnfctr 24447 xmetresbl 24448 xrsblre 24834 ecxpid 33390 lsmsnorb 33420 ecin0 38354 eqvrelth 38613 qsalrel 42281 | 
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