Theorem relsnop 5742

Description: A singleton of an ordered pair is a relation. (Contributed by NM, 17-May-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
relsn.1	⊢ 𝐴 ∈ V
relsnop.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
relsnop	⊢ Rel {⟨𝐴, 𝐵⟩}

Proof of Theorem relsnop

Step	Hyp	Ref	Expression
1		relsn.1	. 2 ⊢ 𝐴 ∈ V
2		relsnop.2	. 2 ⊢ 𝐵 ∈ V
3		relsnopg 5740	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Rel {⟨𝐴, 𝐵⟩})
4	1, 2, 3	mp2an 692	1 ⊢ Rel {⟨𝐴, 𝐵⟩}

Syntax hints:   ∈ wcel 2109  Vcvv 3433  {csn 4573  ⟨cop 4579  Rel wrel 5618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pr 5367

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-opab 5151  df-xp 5619  df-rel 5620

This theorem is referenced by:  fsn  7062  imasaddfnlem  17419  ex-res  30372  0funcg  49084  0funcALT  49087  functermc2  49508