Theorem relsng 5638

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 5526	. 2 ⊢ (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2		snssg 4678	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
3	1, 2	bitr4id 293	1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  {csn 4525   × cxp 5517  Rel wrel 5524

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-sn 4526  df-rel 5526

This theorem is referenced by:  relsnb  5639  relsnopg  5640  relsn  5641  relsn2  6036