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 4250. (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 2610 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix3i 1228 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴) |
3 | eltpg 4174 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 247 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 {ctp 4129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 df-tp 4130 |
This theorem is referenced by: tpid3 4250 tpnzd 4257 f1dom3fv3dif 6425 f1dom3el3dif 6426 en3lplem1 8394 en3lp 8396 nb3graprlem1 25980 en3lplem1VD 38100 en3lpVD 38102 etransclem48 39175 nb3grprlem1 40608 cplgr3v 40657 |
Copyright terms: Public domain | W3C validator |