Theorem relsn 5804

Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Hypothesis

Ref	Expression
relsn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
relsn	⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Proof of Theorem relsn

Step	Hyp	Ref	Expression
1		relsn.1	. 2 ⊢ 𝐴 ∈ V
2		relsng 5801	. 2 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
3	1, 2	ax-mp 5	1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Syntax hints:   ↔ wb 205   ∈ wcel 2106  Vcvv 3474  {csn 4628   × cxp 5674  Rel wrel 5681

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-sn 4629  df-rel 5683

This theorem is referenced by:  setscom  17112  setsid  17140