Theorem relsn2m 4734

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 4386	. 2 ⊢ (Rel {A} ↔ A ∈ (V × V))
3		dmsnm 4729	. 2 ⊢ (A ∈ (V × V) ↔ ∃x x ∈ dom {A})
4	2, 3	bitri 173	1 ⊢ (Rel {A} ↔ ∃x x ∈ dom {A})

Syntax hints:   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  Vcvv 2551  {csn 3367   × cxp 4286  dom cdm 4288  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298

This theorem is referenced by: (None)