Theorem relsnop 4387

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 4333	. 2 ⊢ ⟨A, B⟩ ∈ (V × V)
4		opexgOLD 3956	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ⟨A, B⟩ ∈ V)
5	1, 2, 4	mp2an 402	. . 3 ⊢ ⟨A, B⟩ ∈ V
6	5	relsn 4386	. 2 ⊢ (Rel {⟨A, B⟩} ↔ ⟨A, B⟩ ∈ (V × V))
7	3, 6	mpbir 134	1 ⊢ Rel {⟨A, B⟩}

Syntax hints:   ∈ wcel 1390  Vcvv 2551  {csn 3367  ⟨cop 3370   × cxp 4286  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-rel 4295

This theorem is referenced by:  cnvsn  4746  fsn  5278