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Theorem relsng 4688
 Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
relsng (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))

Proof of Theorem relsng
StepHypRef Expression
1 df-rel 4592 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 snssg 3692 . 2 (𝐴𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
31, 2bitr4id 198 1 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2128  Vcvv 2712   ⊆ wss 3102  {csn 3560   × cxp 4583  Rel wrel 4590 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-sn 3566  df-rel 4592 This theorem is referenced by:  relsnopg  4689  setscom  12201  setsslid  12211
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