Theorem relsn 5763

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 5760	. 2 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
3	1, 2	ax-mp 5	1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3442  {csn 4582   × cxp 5632  Rel wrel 5639

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 3444  df-ss 3920  df-sn 4583  df-rel 5641

This theorem is referenced by:  setscom  17121  setsid  17148