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Mirrors > Home > ILE Home > Th. List > relsnop | GIF version |
Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
relsn.1 | ⊢ 𝐴 ∈ V |
relsnop.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
relsnop | ⊢ Rel {〈𝐴, 𝐵〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relsn.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | relsnop.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opelvv 4408 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
4 | 1, 2 | opex 3984 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V |
5 | 4 | relsn 4461 | . 2 ⊢ (Rel {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝐵〉 ∈ (V × V)) |
6 | 3, 5 | mpbir 144 | 1 ⊢ Rel {〈𝐴, 𝐵〉} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 Vcvv 2601 {csn 3398 〈cop 3401 × cxp 4361 Rel wrel 4368 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 df-rel 4370 |
This theorem is referenced by: cnvsn 4823 fsn 5356 |
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