Theorem relsng 4714

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	⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))

Proof of Theorem relsng

Step	Hyp	Ref	Expression
1		df-rel 4618	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		snssg 3716	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
3	1, 2	bitr4id 198	1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))

Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  {csn 3583   × cxp 4609  Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-rel 4618

This theorem is referenced by:  relsnopg  4715  setscom  12456  setsslid  12466