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Mirrors > Home > ILE Home > Th. List > tpid2g | GIF version |
Description: Closed theorem form of tpid2 3683. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
tpid2g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2164 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix2i 1159 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷) |
3 | eltpg 3615 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶, 𝐴, 𝐷} ↔ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷))) | |
4 | 2, 3 | mpbiri 167 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 966 = wceq 1342 ∈ wcel 2135 {ctp 3572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-tp 3578 |
This theorem is referenced by: rngplusgg 12454 srngplusgd 12461 lmodplusgd 12472 ipsaddgd 12480 ipsvscad 12483 topgrpplusgd 12490 |
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