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 4709. (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 2821 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix3i 1331 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴) |
3 | eltpg 4623 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐷, 𝐴} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 260 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 {ctp 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-sn 4568 df-pr 4570 df-tp 4572 |
This theorem is referenced by: tpid3 4709 f1dom3fv3dif 7026 f1dom3el3dif 7027 en3lplem1 9075 en3lp 9077 nb3grprlem1 27162 cplgr3v 27217 cshw1s2 30634 cyc3co2 30782 en3lplem1VD 41197 en3lpVD 41199 limsupequzlem 42023 etransclem48 42587 |
Copyright terms: Public domain | W3C validator |