Theorem relsnop 5744

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 5742	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Rel {⟨𝐴, 𝐵⟩})
4	1, 2, 3	mp2an 692	1 ⊢ Rel {⟨𝐴, 𝐵⟩}

Syntax hints:   ∈ wcel 2111  Vcvv 3436  {csn 4573  ⟨cop 4579  Rel wrel 5619

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-opab 5152  df-xp 5620  df-rel 5621

This theorem is referenced by:  fsn  7068  imasaddfnlem  17432  ex-res  30421  0funcg  49196  0funcALT  49199  functermc2  49620