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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 2734 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | 3mix1i 1332 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶) |
3 | tpid1.1 | . . 3 ⊢ 𝐴 ∈ V | |
4 | 3 | eltp 4693 | . 2 ⊢ (𝐴 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵 ∨ 𝐴 = 𝐶)) |
5 | 2, 4 | mpbir 231 | 1 ⊢ 𝐴 ∈ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1085 = wceq 1536 ∈ wcel 2105 Vcvv 3477 {ctp 4634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-un 3967 df-sn 4631 df-pr 4633 df-tp 4635 |
This theorem is referenced by: tpnz 4783 hash3tpb 14530 wrdl3s3 14997 cffldtocusgr 29478 cffldtocusgrOLD 29479 umgrwwlks2on 29986 s3rnOLD 32914 cyc3evpm 33152 sgnsf 33164 sgncl 34519 prodfzo03 34596 circlevma 34635 circlemethhgt 34636 hgt750lemg 34647 hgt750lemb 34649 hgt750lema 34650 hgt750leme 34651 tgoldbachgtde 34653 tgoldbachgt 34656 kur14lem7 35196 kur14lem9 35198 brtpid1 35700 rabren3dioph 42802 fourierdlem102 46163 fourierdlem114 46175 etransclem48 46237 usgrexmpl1tri 47919 usgrexmpl2nb0 47925 usgrexmpl2nb1 47926 usgrexmpl2nb2 47927 usgrexmpl2nb3 47928 usgrexmpl2nb4 47929 usgrexmpl2nb5 47930 |
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