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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 2798 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix1i 1330 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶) |
3 | tpid1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | eltp 4586 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
5 | 2, 4 | mpbir 234 | 1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {ctp 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 df-sn 4526 df-pr 4528 df-tp 4530 |
This theorem is referenced by: tpnz 4675 wrdl3s3 14317 cffldtocusgr 27237 umgrwwlks2on 27743 s3rn 30648 cyc3evpm 30842 sgnsf 30854 sgncl 31906 prodfzo03 31984 circlevma 32023 circlemethhgt 32024 hgt750lemg 32035 hgt750lemb 32037 hgt750lema 32038 hgt750leme 32039 tgoldbachgtde 32041 tgoldbachgt 32044 kur14lem7 32572 kur14lem9 32574 brtpid1 33064 rabren3dioph 39756 fourierdlem102 42850 fourierdlem114 42862 etransclem48 42924 |
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