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Theorem relsn2 6216
Description: A singleton is a relation iff it has a nonempty domain. (Contributed by NM, 25-Sep-2013.) Make hypothesis an antecedent. (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
relsn2 (𝐴𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))

Proof of Theorem relsn2
StepHypRef Expression
1 relsng 5803 . 2 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2 dmsnn0 6211 . 2 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
31, 2bitrdi 287 1 (𝐴𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wne 2937  Vcvv 3471  c0 4323  {csn 4629   × cxp 5676  dom cdm 5678  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-dm 5688
This theorem is referenced by: (None)
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