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Theorem relsnop 5135
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
StepHypRef Expression
1 relsn.1 . . 3 𝐴 ∈ V
2 relsnop.2 . . 3 𝐵 ∈ V
31, 2opelvv 5077 . 2 𝐴, 𝐵⟩ ∈ (V × V)
4 opex 4852 . . 3 𝐴, 𝐵⟩ ∈ V
54relsn 5134 . 2 (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V))
63, 5mpbir 219 1 Rel {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:  wcel 1976  Vcvv 3172  {csn 4124  cop 4130   × cxp 5025  Rel wrel 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5033  df-rel 5034
This theorem is referenced by:  cnvsn  5521  fsn  6292  imasaddfnlem  15959  ex-res  26483
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