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Mirrors > Home > ILE Home > Th. List > eltp | GIF version |
Description: A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
eltp.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eltp | ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eltp.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eltpg 3416 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷))) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ∈ {𝐵, 𝐶, 𝐷} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ∨ 𝐴 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∨ w3o 884 = wceq 1243 ∈ wcel 1393 Vcvv 2557 {ctp 3377 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-tp 3383 |
This theorem is referenced by: dftp2 3419 tpid1 3481 tpid2 3482 tpid3 3484 |
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