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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 5410 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) | |
2 | 1 | ancoms 468 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
3 | df-ec 7631 | . . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 3 | eleq2i 2680 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵})) |
5 | df-br 4584 | . 2 ⊢ (𝐵𝑅𝐴 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | |
6 | 2, 4, 5 | 3bitr4g 302 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 {csn 4125 〈cop 4131 class class class wbr 4583 “ cima 5041 [cec 7627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 |
This theorem is referenced by: elec 7673 relelec 7674 ecdmn0 7676 erth 7678 erdisj 7681 qsel 7713 orbsta 17569 sylow2alem1 17855 sylow2blem1 17858 sylow3lem3 17867 efgi2 17961 tgpconcompeqg 21725 xmetec 22049 blpnfctr 22051 xmetresbl 22052 xrsblre 22422 |
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