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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 3689 | . 2 ⊢ 𝐵 ∈ {𝐵, 𝐴} |
3 | prcom 3659 | . 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵} | |
4 | 2, 3 | eleqtri 2245 | 1 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 Vcvv 2730 {cpr 3584 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 |
This theorem is referenced by: prel12 3758 opi2 4218 opeluu 4435 ontr2exmid 4509 onsucelsucexmid 4514 regexmidlemm 4516 ordtri2or2exmid 4555 ontri2orexmidim 4556 dmrnssfld 4874 funopg 5232 acexmidlema 5844 acexmidlemcase 5848 acexmidlem2 5850 1lt2o 6421 2dom 6783 unfiexmid 6895 djuss 7047 exmidonfinlem 7170 exmidfodomrlemr 7179 exmidfodomrlemrALT 7180 exmidaclem 7185 cnelprrecn 7910 mnfxr 7976 sup3exmid 8873 m1expcl2 10498 lgsdir2lem3 13725 bdop 13910 2o01f 14029 iswomni0 14083 |
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