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Mirrors > Home > MPE Home > Th. List > tpid2g | Structured version Visualization version GIF version |
Description: Closed theorem form of tpid2 4717. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
tpid2g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix2i 1333 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷) |
3 | eltpg 4632 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷))) | |
4 | 2, 3 | mpbiri 257 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1085 = wceq 1540 ∈ wcel 2105 {ctp 4576 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3443 df-un 3902 df-sn 4573 df-pr 4575 df-tp 4577 |
This theorem is referenced by: cplgr3v 28032 cshw1s2 31460 cyc3co2 31635 limsupequzlem 43588 |
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