Theorem relsn2m 5238

Description: A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)

Hypothesis

Ref	Expression
relsn2m.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
relsn2m	⊢ (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴})

Distinct variable group:   𝑥,𝐴

Proof of Theorem relsn2m

Step	Hyp	Ref	Expression
1		relsn2m.1	. . 3 ⊢ 𝐴 ∈ V
2	1	relsn 4860	. 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3		dmsnm 5233	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
4	2, 3	bitri 184	1 ⊢ (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴})

Syntax hints:   ↔ wb 105  ∃wex 1541   ∈ wcel 2205  Vcvv 2815  {csn 3694   × cxp 4752  dom cdm 4754  Rel wrel 4759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-dm 4764

This theorem is referenced by: (None)