Theorem relsnop 4830

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	. . 3 ⊢ 𝐴 ∈ V
2		relsnop.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	opelvv 4774	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
4	1, 2	opex 4319	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5	4	relsn 4829	. 2 ⊢ (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V))
6	3, 5	mpbir 146	1 ⊢ Rel {⟨𝐴, 𝐵⟩}

Syntax hints:   ∈ wcel 2200  Vcvv 2800  {csn 3667  ⟨cop 3670   × cxp 4721  Rel wrel 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-xp 4729  df-rel 4730

This theorem is referenced by:  cnvsn  5217  fsn  5815