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Mirrors > Home > ILE Home > Th. List > relsng | GIF version |
Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.) |
Ref | Expression |
---|---|
relsng | ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssg 3626 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))) | |
2 | df-rel 4516 | . 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V)) | |
3 | 1, 2 | syl6rbbr 198 | 1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1465 Vcvv 2660 ⊆ wss 3041 {csn 3497 × cxp 4507 Rel wrel 4514 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 df-ss 3054 df-sn 3503 df-rel 4516 |
This theorem is referenced by: relsnopg 4613 setscom 11926 setsslid 11936 |
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