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Theorem relsn2 5861
 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 5474 . 2 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
2 dmsnn0 5856 . 2 (𝐴 ∈ (V × V) ↔ dom {𝐴} ≠ ∅)
31, 2syl6bb 279 1 (𝐴𝑉 → (Rel {𝐴} ↔ dom {𝐴} ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∈ wcel 2107   ≠ wne 2969  Vcvv 3398  ∅c0 4141  {csn 4398   × cxp 5355  dom cdm 5357  Rel wrel 5362 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-dm 5367 This theorem is referenced by: (None)
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