Theorem relsn2m 5140

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 4768	. 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3		dmsnm 5135	. 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 1506   ∈ wcel 2167  Vcvv 2763  {csn 3622   × cxp 4661  dom cdm 4663  Rel wrel 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-dm 4673

This theorem is referenced by: (None)