![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4690 | . 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 1085 = wceq 1536 ∈ wcel 2105 {ctp 4634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-un 3967 df-sn 4631 df-pr 4633 df-tp 4635 |
This theorem is referenced by: fvf1tp 13825 tpfo 14535 prm23lt5 16847 perfectlem2 27288 zabsle1 27354 gsumtp 33043 cyc3co2 33142 sgnmulsgn 34530 sgnmulsgp 34531 kur14lem7 35196 omcl3g 43323 fmtnofz04prm 47501 perfectALTVlem2 47646 |
Copyright terms: Public domain | W3C validator |