Theorem relsn2 5605

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 5223	. 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3		dmsnn0 5600	. 2 ⊢ (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
4	2, 3	bitri 264	1 ⊢ (Rel {𝐴} ↔ dom {𝐴} ≠ ∅)

Syntax hints:   ↔ wb 196   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200  ∅c0 3915  {csn 4177   × cxp 5112  dom cdm 5114  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124

This theorem is referenced by: (None)