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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 6093 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) | |
2 | 1 | ancoms 457 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ (𝑅 “ {𝐵}) ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) |
3 | df-ec 8727 | . . 3 ⊢ [𝐵]𝑅 = (𝑅 “ {𝐵}) | |
4 | 3 | eleq2i 2817 | . 2 ⊢ (𝐴 ∈ [𝐵]𝑅 ↔ 𝐴 ∈ (𝑅 “ {𝐵})) |
5 | df-br 5150 | . 2 ⊢ (𝐵𝑅𝐴 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | |
6 | 2, 4, 5 | 3bitr4g 313 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 {csn 4630 〈cop 4636 class class class wbr 5149 “ cima 5681 [cec 8723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 |
This theorem is referenced by: ecref 8769 elec 8770 relelec 8771 ecdmn0 8773 erth 8775 erdisj 8778 qsel 8815 ghmqusnsglem1 19243 ghmquskerlem1 19246 orbsta 19276 sylow2alem1 19584 sylow2blem1 19587 sylow3lem3 19596 efgi2 19692 rngqiprngfulem2 21219 rngqipring1 21223 tgpconncompeqg 24060 xmetec 24384 blpnfctr 24386 xmetresbl 24387 xrsblre 24771 ecxpid 33172 lsmsnorb 33203 ecin0 37954 eqvrelth 38213 qsalrel 41864 |
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