Theorem relsnopg 5785

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 5696	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2		opex 5439	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
3		relsng 5783	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V)))
4	2, 3	mp1i 14	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V)))
5	1, 4	mpbird 260	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Rel {⟨𝐴, 𝐵⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Vcvv 3450  {csn 4584  ⟨cop 4590   × cxp 5653  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5661  df-rel 5662

This theorem is used by:  relsnop  5787  cnvsng  6221