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Theorem relsnopg 5757
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 5671 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2 opex 5419 . . 3 𝐴, 𝐵⟩ ∈ V
3 relsng 5755 . . 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 256 1 ((𝐴𝑉𝐵𝑊) → Rel {⟨𝐴, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3443  {csn 4584  cop 4590   × cxp 5629  Rel wrel 5636
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5166  df-xp 5637  df-rel 5638
This theorem is referenced by:  relsnop  5759  cnvsng  6173
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