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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 3687 | . 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 2141 Vcvv 2730 {cpr 3584 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 |
This theorem is referenced by: prid2 3690 prnz 3705 preqr1 3755 preq12b 3757 prel12 3758 opi1 4217 opeluu 4435 onsucelsucexmidlem1 4512 regexmidlem1 4517 reg2exmidlema 4518 opthreg 4540 ordtri2or2exmid 4555 ontri2orexmidim 4556 dmrnssfld 4874 funopg 5232 acexmidlemb 5845 0lt2o 6420 2dom 6783 unfiexmid 6895 djuss 7047 exmidomni 7118 exmidonfinlem 7170 exmidaclem 7185 reelprrecn 7909 pnfxr 7972 sup3exmid 8873 lgsdir2lem3 13725 bdop 13910 2o01f 14029 iswomni0 14083 |
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