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Theorem relsnopg 5643
 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 5562 . 2 ((𝐴𝑉𝐵𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2 opex 5322 . . 3 𝐴, 𝐵⟩ ∈ V
3 relsng 5641 . . 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 260 1 ((𝐴𝑉𝐵𝑊) → Rel {⟨𝐴, 𝐵⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2112  Vcvv 3410  {csn 4520  ⟨cop 4526   × cxp 5520  Rel wrel 5527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730  ax-sep 5167  ax-nul 5174  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rex 3077  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4419  df-sn 4521  df-pr 4523  df-op 4527  df-opab 5093  df-xp 5528  df-rel 5529 This theorem is referenced by:  relsnop  5645  cnvsng  6050
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