Theorem relsn2m 5137

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 4765	. 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3		dmsnm 5132	. 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 1503   ∈ wcel 2164  Vcvv 2760  {csn 3619   × cxp 4658  dom cdm 4660  Rel wrel 4665

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-dm 4670

This theorem is referenced by: (None)