Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3665 | . 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 2128 Vcvv 2712 {cpr 3562 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-sn 3567 df-pr 3568 |
This theorem is referenced by: prid2 3668 prnz 3683 preqr1 3733 preq12b 3735 prel12 3736 opi1 4194 opeluu 4412 onsucelsucexmidlem1 4489 regexmidlem1 4494 reg2exmidlema 4495 opthreg 4517 ordtri2or2exmid 4532 ontri2orexmidim 4533 dmrnssfld 4851 funopg 5206 acexmidlemb 5818 0lt2o 6390 2dom 6752 unfiexmid 6864 djuss 7016 exmidomni 7087 exmidonfinlem 7130 exmidaclem 7145 reelprrecn 7869 pnfxr 7932 sup3exmid 8833 bdop 13521 2o01f 13639 iswomni0 13693 |
Copyright terms: Public domain | W3C validator |