Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2137 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix1i 1153 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶) |
3 | tpid1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | eltp 3566 | . 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 961 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {ctp 3524 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-tp 3530 |
This theorem is referenced by: tpnz 3643 |
Copyright terms: Public domain | W3C validator |