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| Mirrors > Home > MPE Home > Th. List > eltpi | Structured version Visualization version GIF version | ||
| Description: A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| Ref | Expression |
|---|---|
| eltpi | ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eltpg 4686 | . 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) | |
| 2 | 1 | ibi 267 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 {ctp 4630 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 df-tp 4631 |
| This theorem is referenced by: fvf1tp 13829 tpfo 14539 prm23lt5 16852 perfectlem2 27274 zabsle1 27340 gsumtp 33061 cyc3co2 33160 sgnmulsgn 34552 sgnmulsgp 34553 kur14lem7 35217 omcl3g 43347 fmtnofz04prm 47564 perfectALTVlem2 47709 |
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