Theorem relsn2 6185

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 5764	. 2 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2		dmsnn0 6180	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
3	1, 2	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109   ≠ wne 2925  Vcvv 3447  ∅c0 4296  {csn 4589   × cxp 5636  dom cdm 5638  Rel wrel 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648

This theorem is referenced by: (None)