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Theorem relsn2m 4896
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 4539 . 2 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
3 dmsnm 4891 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
42, 3bitri 182 1 (Rel {𝐴} ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
Colors of variables: wff set class
Syntax hints:  wb 103  wex 1426  wcel 1438  Vcvv 2619  {csn 3444   × cxp 4434  dom cdm 4436  Rel wrel 4441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-br 3844  df-opab 3898  df-xp 4442  df-rel 4443  df-dm 4446
This theorem is referenced by: (None)
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