Theorem relsnop 5642

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 5640	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Rel {⟨𝐴, 𝐵⟩})
4	1, 2, 3	mp2an 691	1 ⊢ Rel {⟨𝐴, 𝐵⟩}

Syntax hints:   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526

This theorem is referenced by:  fsn  6874  imasaddfnlem  16793  ex-res  28226