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Mirrors > Home > ILE Home > Th. List > prid1 | GIF version |
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
prid1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
prid1 | ⊢ 𝐴 ∈ {𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid1.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | prid1g 3496 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵}) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 Vcvv 2601 {cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: prid2 3499 prnz 3512 preqr1 3560 preq12b 3562 prel12 3563 opi1 3987 opeluu 4200 onsucelsucexmidlem1 4271 regexmidlem1 4276 reg2exmidlema 4277 opthreg 4299 ordtri2or2exmid 4314 dmrnssfld 4613 funopg 4954 acexmidlemb 5524 2dom 6308 reelprrecn 7108 pnfxr 8846 bdop 10666 |
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