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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 4646 | . 2 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) | |
2 | 1 | ibi 266 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 {ctp 4590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3447 df-un 3915 df-sn 4587 df-pr 4589 df-tp 4591 |
This theorem is referenced by: prm23lt5 16686 perfectlem2 26578 zabsle1 26644 cyc3co2 31989 sgnmulsgn 33149 sgnmulsgp 33150 kur14lem7 33806 omcl3g 41653 fmtnofz04prm 45759 perfectALTVlem2 45904 |
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