Theorem relsn2m 5256

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 4878	. 2 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3		dmsnm 5251	. 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 1545   ∈ wcel 2209  Vcvv 2821  {csn 3708   × cxp 4770  dom cdm 4772  Rel wrel 4777

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-rel 4779  df-dm 4782

This theorem is referenced by: (None)