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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 3599 | . 2 ⊢ 𝐵 ∈ {𝐵, 𝐴} |
3 | prcom 3569 | . 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵} | |
4 | 2, 3 | eleqtri 2192 | 1 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 Vcvv 2660 {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: prel12 3668 opi2 4125 opeluu 4341 ontr2exmid 4410 onsucelsucexmid 4415 regexmidlemm 4417 ordtri2or2exmid 4456 dmrnssfld 4772 funopg 5127 acexmidlema 5733 acexmidlemcase 5737 acexmidlem2 5739 1lt2o 6307 2dom 6667 unfiexmid 6774 djuss 6923 exmidonfinlem 7017 exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 exmidaclem 7032 cnelprrecn 7724 mnfxr 7790 sup3exmid 8683 m1expcl2 10283 bdop 13000 isomninnlem 13152 |
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