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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elecALTV | Structured version Visualization version GIF version | ||
| Description: Elementhood in the 𝑅-coset of 𝐴. Theorem 72 of [Suppes] p. 82. (I think we should replace elecg 8790 with this original form of Suppes. Peter Mazsa). (Contributed by Mario Carneiro, 9-Jul-2014.) | 
| Ref | Expression | 
|---|---|
| elecALTV | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elimasng 6106 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ 𝑅)) | |
| 2 | df-ec 8748 | . . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
| 3 | 2 | eleq2i 2832 | . 2 ⊢ (𝐵 ∈ [𝐴]𝑅 ↔ 𝐵 ∈ (𝑅 “ {𝐴})) | 
| 4 | df-br 5143 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 5 | 1, 3, 4 | 3bitr4g 314 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐵)) | 
| 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: eldm4 38276 exan3 38296 exanres3 38298 ecin0 38354 dfcoss2 38415 eldm1cossres2 38463 eqvrelth 38613 eqvreldisj 38616 eqvrelqsel 38618 erimeq2 38680 disjlem19 38803 | 
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