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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 3635 | . 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ {𝐴, 𝐵}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 𝐴 ∈ {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 Vcvv 2689 {cpr 3533 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 |
This theorem is referenced by: prid2 3638 prnz 3653 preqr1 3703 preq12b 3705 prel12 3706 opi1 4162 opeluu 4379 onsucelsucexmidlem1 4451 regexmidlem1 4456 reg2exmidlema 4457 opthreg 4479 ordtri2or2exmid 4494 dmrnssfld 4810 funopg 5165 acexmidlemb 5774 0lt2o 6346 2dom 6707 unfiexmid 6814 djuss 6963 exmidomni 7022 exmidonfinlem 7066 exmidaclem 7081 reelprrecn 7779 pnfxr 7842 sup3exmid 8739 bdop 13244 2o01f 13364 |
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