![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tpid3g | Structured version Visualization version GIF version |
Description: Closed theorem form of tpid3 4778. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 30-Apr-2021.) |
Ref | Expression |
---|---|
tpid3g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix3i 1334 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴) |
3 | eltpg 4691 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1085 = wceq 1537 ∈ wcel 2106 {ctp 4635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 df-tp 4636 |
This theorem is referenced by: tpid3 4778 f1dom3fv3dif 7288 f1dom3el3dif 7289 en3lplem1 9650 en3lp 9652 tpf 14535 nb3grprlem1 29412 cplgr3v 29467 cshw1s2 32930 cyc3co2 33143 en3lplem1VD 44841 en3lpVD 44843 limsupequzlem 45678 etransclem48 46238 |
Copyright terms: Public domain | W3C validator |