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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 2117 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | orci 705 | . 2 ⊢ (𝐴 = 𝐴 ∨ 𝐴 = 𝐵) |
3 | elprg 3517 | . 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 682 = wceq 1316 ∈ wcel 1465 {cpr 3498 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-sn 3503 df-pr 3504 |
This theorem is referenced by: prid2g 3598 prid1 3599 preqr1g 3663 opth1 4128 en2lp 4439 acexmidlemcase 5737 en2eqpr 6769 m1expcl2 10283 maxabslemval 10948 xrmaxiflemval 10987 xrmaxaddlem 10997 2strbasg 11987 2strbas1g 11990 coseq0negpitopi 12844 |
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