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Mirrors > Home > ILE Home > Th. List > tpid3 | GIF version |
Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
tpid3.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
tpid3 | ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2082 | . . 3 ⊢ 𝐶 = 𝐶 | |
2 | 1 | 3mix3i 1113 | . 2 ⊢ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶) |
3 | tpid3.1 | . . 3 ⊢ 𝐶 ∈ V | |
4 | 3 | eltp 3448 | . 2 ⊢ (𝐶 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ∨ 𝐶 = 𝐶)) |
5 | 2, 4 | mpbir 144 | 1 ⊢ 𝐶 ∈ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∨ w3o 919 = wceq 1285 ∈ wcel 1434 Vcvv 2602 {ctp 3408 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-v 2604 df-un 2978 df-sn 3412 df-pr 3413 df-tp 3414 |
This theorem is referenced by: (None) |
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