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Mirrors > Home > ILE Home > Th. List > tpid2g | GIF version |
Description: Closed theorem form of tpid2 3583. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
tpid2g | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐴, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2100 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix2i 1122 | . 2 ⊢ (𝐴 = 𝐶 ∨ 𝐴 = 𝐴 ∨ 𝐴 = 𝐷) |
3 | eltpg 3516 | . 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 929 = wceq 1299 ∈ wcel 1448 {ctp 3476 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 df-tp 3482 |
This theorem is referenced by: rngplusgg 11858 srngplusgd 11865 lmodplusgd 11876 ipsaddgd 11884 ipsvscad 11887 topgrpplusgd 11894 |
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