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Mirrors > Home > ILE Home > Th. List > prid2 | GIF version |
Description: An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
prid2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
prid2 | ⊢ 𝐵 ∈ {𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prid2.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | 1 | prid1 3665 | . 2 ⊢ 𝐵 ∈ {𝐵, 𝐴} |
3 | prcom 3635 | . 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵} | |
4 | 2, 3 | eleqtri 2232 | 1 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2128 Vcvv 2712 {cpr 3561 |
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 3566 df-pr 3567 |
This theorem is referenced by: prel12 3734 opi2 4193 opeluu 4410 ontr2exmid 4484 onsucelsucexmid 4489 regexmidlemm 4491 ordtri2or2exmid 4530 ontri2orexmidim 4531 dmrnssfld 4849 funopg 5204 acexmidlema 5815 acexmidlemcase 5819 acexmidlem2 5821 1lt2o 6389 2dom 6750 unfiexmid 6862 djuss 7014 exmidonfinlem 7128 exmidfodomrlemr 7137 exmidfodomrlemrALT 7138 exmidaclem 7143 cnelprrecn 7868 mnfxr 7934 sup3exmid 8828 m1expcl2 10441 bdop 13461 2o01f 13579 iswomni0 13633 |
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