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Theorem relsng 5743
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 5627 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 snssg 4731 . 2 (𝐴𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
31, 2bitr4id 289 1 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2105  Vcvv 3441  wss 3898  {csn 4573   × cxp 5618  Rel wrel 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-in 3905  df-ss 3915  df-sn 4574  df-rel 5627
This theorem is referenced by:  relsnb  5744  relsnopg  5745  relsn  5746  relsn2  6150
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