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Mirrors > Home > MPE Home > Th. List > tpid2g | Structured version Visualization version GIF version |
Description: Closed theorem form of tpid2 4698. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
tpid2g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix2i 1326 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷) |
3 | eltpg 4615 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷))) | |
4 | 2, 3 | mpbiri 259 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 {ctp 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-un 3938 df-sn 4558 df-pr 4560 df-tp 4562 |
This theorem is referenced by: cplgr3v 27144 cshw1s2 30561 cyc3co2 30709 limsupequzlem 41879 |
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