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Theorem relsng 5686
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 df-rel 5573 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 snssg 4713 . 2 (𝐴𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
31, 2bitr4id 293 1 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2111  Vcvv 3421  wss 3881  {csn 4556   × cxp 5564  Rel wrel 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3423  df-in 3888  df-ss 3898  df-sn 4557  df-rel 5573
This theorem is referenced by:  relsnb  5687  relsnopg  5688  relsn  5689  relsn2  6090
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