Theorem relsn2 5523

Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.)

Hypothesis

Ref	Expression
relsn2.1	⊢ 𝐴 ∈ V

Assertion

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

Proof of Theorem relsn2

Step	Hyp	Ref	Expression
1		relsn2.1	. . 3 ⊢ 𝐴 ∈ V
2	1	relsn 5146	. 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3		dmsnn0 5518	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
4	2, 3	bitri 263	1 ⊢ (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 195   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  {csn 4125   × cxp 5036  dom cdm 5038  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048

This theorem is referenced by: (None)