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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 4628 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
2 | opexg 4183 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
3 | relsng 4682 | . . 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 166 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {〈𝐴, 𝐵〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2125 Vcvv 2709 {csn 3556 〈cop 3559 × cxp 4577 Rel wrel 4584 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-rex 2438 df-v 2711 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-opab 4022 df-xp 4585 df-rel 4586 |
This theorem is referenced by: (None) |
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