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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
StepHypRef Expression
1 relsn.1 . . 3 A V
2 relsnop.2 . . 3 B V
31, 2opelvv 4333 . 2 A, B (V × V)
4 opexgOLD 3956 . . . 4 ((A V B V) → ⟨A, B V)
51, 2, 4mp2an 402 . . 3 A, B V
65relsn 4386 . 2 (Rel {⟨A, B⟩} ↔ ⟨A, B (V × V))
73, 6mpbir 134 1 Rel {⟨A, B⟩}
Colors of variables: wff set class
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-bnd 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
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