Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  relsn2m GIF version

Theorem relsn2m 5009
 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 4644 . 2 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3 dmsnm 5004 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
42, 3bitri 183 1 (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  {csn 3527   × cxp 4537  dom cdm 4539  Rel wrel 4544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-dm 4549 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator