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Mirrors > Home > MPE Home > Th. List > tpid1 | Structured version Visualization version 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 2610 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix1i 1226 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶) |
3 | tpid1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | eltp 4177 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
5 | 2, 4 | mpbir 220 | 1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {ctp 4129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 df-sn 4126 df-pr 4128 df-tp 4130 |
This theorem is referenced by: tpnz 4256 wrdl3s3 13553 2pthlem2 26126 usgra2adedgwlkonALT 26144 sgnsf 29060 sgncl 29927 kur14lem7 30448 kur14lem9 30450 brtpid1 30857 rabren3dioph 36397 fourierdlem102 39101 fourierdlem114 39113 etransclem48 39175 umgrwwlks2on 41161 |
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