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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
StepHypRef Expression
1 relsn2m.1 . . 3 A V
21relsn 4386 . 2 (Rel {A} ↔ A (V × V))
3 dmsnm 4729 . 2 (A (V × V) ↔ x x dom {A})
42, 3bitri 173 1 (Rel {A} ↔ x x dom {A})
 Colors of variables: wff set class 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)
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