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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 8321 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 5948 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝑅 “ {𝐴}) ↔ 〈𝐴, 𝐵〉 ∈ 𝑅)) | |
2 | df-ec 8280 | . . 3 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | |
3 | 2 | eleq2i 2901 | . 2 ⊢ (𝐵 ∈ [𝐴]𝑅 ↔ 𝐵 ∈ (𝑅 “ {𝐴})) |
4 | df-br 5058 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
5 | 1, 3, 4 | 3bitr4g 315 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑅 ↔ 𝐴𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 {csn 4557 〈cop 4563 class class class wbr 5057 “ cima 5551 [cec 8276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ec 8280 |
This theorem is referenced by: eldm4 35412 exan3 35432 exanres3 35434 ecin0 35487 dfcoss2 35541 eldm1cossres2 35581 eqvrelth 35726 eqvreldisj 35729 eqvrelqsel 35731 erim2 35791 |
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