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Mirrors > Home > ILE Home > Th. List > prid1g | GIF version |
Description: An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.) |
Ref | Expression |
---|---|
prid1g | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2115 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | orci 703 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵) |
3 | elprg 3513 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐴, 𝐵} ↔ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵))) | |
4 | 2, 3 | mpbiri 167 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 680 = wceq 1314 ∈ wcel 1463 {cpr 3494 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 |
This theorem is referenced by: prid2g 3594 prid1 3595 preqr1g 3659 opth1 4118 en2lp 4429 acexmidlemcase 5723 en2eqpr 6754 m1expcl2 10205 maxabslemval 10869 xrmaxiflemval 10908 xrmaxaddlem 10918 2strbasg 11900 2strbas1g 11903 |
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