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Mirrors > Home > ILE Home > Th. List > tpid1 | 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 |
---|---|
tpid1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
tpid1 | ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2164 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix1i 1158 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶) |
3 | tpid1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | eltp 3618 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
5 | 2, 4 | mpbir 145 | 1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∨ w3o 966 = wceq 1342 ∈ wcel 2135 Vcvv 2721 {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: tpnz 3695 |
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