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Mirrors > Home > MPE Home > Th. List > tpid2g | Structured version Visualization version GIF version |
Description: Closed theorem form of tpid2 4795. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
tpid2g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix2i 1334 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷) |
3 | eltpg 4709 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷))) | |
4 | 2, 3 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1086 = wceq 1537 ∈ wcel 2103 {ctp 4652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3484 df-un 3975 df-sn 4649 df-pr 4651 df-tp 4653 |
This theorem is referenced by: tpf 14544 cplgr3v 29461 cshw1s2 32919 cyc3co2 33125 limsupequzlem 45577 |
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