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Mirrors > Home > NFE 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 3825 | . 2 ⊢ (A ∈ V → A ∈ {A, B}) | |
3 | 1, 2 | ax-mp 8 | 1 ⊢ A ∈ {A, B} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 Vcvv 2859 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 |
This theorem is referenced by: prid2 3828 prnz 3835 preqr1 4124 preq12b 4127 |
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