Theorem relsng 5814

Description: A singleton is a relation iff it is a singleton on 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 5696	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		snssg 4788	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
3	1, 2	bitr4id 290	1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  {csn 4631   × cxp 5687  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-sn 4632  df-rel 5696

This theorem is referenced by:  relsnb  5815  relsnopg  5816  relsn  5817  relsn2  6234