Theorem relsnop 4770

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 4714	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
4	1, 2	opex 4263	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
5	4	relsn 4769	. 2 ⊢ (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V))
6	3, 5	mpbir 146	1 ⊢ Rel {⟨𝐴, 𝐵⟩}

Syntax hints:   ∈ wcel 2167  Vcvv 2763  {csn 3623  ⟨cop 3626   × cxp 4662  Rel wrel 4669

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-opab 4096  df-xp 4670  df-rel 4671

This theorem is referenced by:  cnvsn  5153  fsn  5737