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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 3516 | . 2 ⊢ 𝐵 ∈ {𝐵, 𝐴} |
3 | prcom 3486 | . 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵} | |
4 | 2, 3 | eleqtri 2157 | 1 ⊢ 𝐵 ∈ {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1434 Vcvv 2610 {cpr 3417 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2612 df-un 2986 df-sn 3422 df-pr 3423 |
This theorem is referenced by: prel12 3583 opi2 4016 opeluu 4228 ontr2exmid 4296 onsucelsucexmid 4301 regexmidlemm 4303 ordtri2or2exmid 4342 dmrnssfld 4643 funopg 4984 acexmidlema 5555 acexmidlemcase 5559 acexmidlem2 5561 2dom 6374 unfiexmid 6463 djuss 6564 cnelprrecn 7241 mnfxr 7307 m1expcl2 9665 bdop 10951 |
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