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Mirrors > Home > ILE Home > Th. List > tpid2 | 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 |
---|---|
tpid2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
tpid2 | ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2115 | . . 3 ⊢ 𝐵 = 𝐵 | |
2 | 1 | 3mix2i 1137 | . 2 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶) |
3 | tpid2.1 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 3 | eltp 3537 | . 2 ⊢ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)) |
5 | 2, 4 | mpbir 145 | 1 ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∨ w3o 944 = wceq 1314 ∈ wcel 1463 Vcvv 2657 {ctp 3495 |
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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 df-tp 3501 |
This theorem is referenced by: (None) |
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