Theorem relsnopg 5750

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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐴, 𝐵⟩})

Proof of Theorem relsnopg

Step	Hyp	Ref	Expression
1		opelvvg 5663	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2		opex 5410	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3		relsng 5748	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V)))
4	2, 3	mp1i 13	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V)))
5	1, 4	mpbird 257	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐴, 𝐵⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  Vcvv 3438  {csn 4578  ⟨cop 4584   × cxp 5620  Rel wrel 5627

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-opab 5159  df-xp 5628  df-rel 5629

This theorem is referenced by:  relsnop  5752  cnvsng  6179