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Mirrors > Home > ILE Home > Th. List > relsnopg | GIF version |
Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.) |
Ref | Expression |
---|---|
relsnopg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {〈𝐴, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelvvg 4487 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
2 | opexg 4055 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | relsng 4541 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ V → (Rel {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝐵〉 ∈ (V × V))) | |
4 | 2, 3 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel {〈𝐴, 𝐵〉} ↔ 〈𝐴, 𝐵〉 ∈ (V × V))) |
5 | 1, 4 | mpbird 165 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {〈𝐴, 𝐵〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1438 Vcvv 2619 {csn 3446 〈cop 3449 × cxp 4436 Rel wrel 4443 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-opab 3900 df-xp 4444 df-rel 4445 |
This theorem is referenced by: (None) |
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