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Theorem relsng 5647
Description: A singleton is a relation iff it is a singleton on an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
relsng (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))

Proof of Theorem relsng
StepHypRef Expression
1 snssg 4690 . 2 (𝐴𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
2 df-rel 5535 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
31, 2syl6rbbr 293 1 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2115  Vcvv 3471  wss 3910  {csn 4540   × cxp 5526  Rel wrel 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-v 3473  df-in 3917  df-ss 3927  df-sn 4541  df-rel 5535
This theorem is referenced by:  relsnb  5648  relsnopg  5649  relsn  5650  relsn2  6042
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