Theorem relsn2 6216

Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.) Make hypothesis an antecedent. (Revised by BJ, 12-Feb-2022.)

Assertion

Ref	Expression
relsn2	⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))

Proof of Theorem relsn2

Step	Hyp	Ref	Expression
1		relsng 5791	. 2 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2		dmsnn0 6211	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3	1, 2	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146   ≠ wne 2961  Vcvv 3458  ∅c0 4289  {csn 4592   × cxp 5662  dom cdm 5664  Rel wrel 5669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5260  ax-pr 5407

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5113  df-opab 5177  df-xp 5670  df-rel 5671  df-dm 5674

This theorem is used by: (None)