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Theorem relsnopg 5816
Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
relsnopg ((𝐴𝑉𝐵𝑊) → Rel {⟨𝐴, 𝐵⟩})

Proof of Theorem relsnopg
StepHypRef Expression
1 opelvvg 5730 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2 opex 5475 . . 3 𝐴, 𝐵⟩ ∈ V
3 relsng 5814 . . 3 (⟨𝐴, 𝐵⟩ ∈ V → (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V)))
42, 3mp1i 13 . 2 ((𝐴𝑉𝐵𝑊) → (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V)))
51, 4mpbird 257 1 ((𝐴𝑉𝐵𝑊) → Rel {⟨𝐴, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  Vcvv 3478  {csn 4631  cop 4637   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  relsnop  5818  cnvsng  6245
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