Theorem relsng 5748

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 5629	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		snssg 4728	. 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 2114  Vcvv 3430   ⊆ wss 3890  {csn 4568   × cxp 5620  Rel wrel 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-ss 3907  df-sn 4569  df-rel 5629

This theorem is referenced by:  relsnb  5749  relsnopg  5750  relsn  5751  relsn2  6168