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Theorem relsnop 4645
 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 4589 . 2 𝐴, 𝐵⟩ ∈ (V × V)
41, 2opex 4151 . . 3 𝐴, 𝐵⟩ ∈ V
54relsn 4644 . 2 (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V))
63, 5mpbir 145 1 Rel {⟨𝐴, 𝐵⟩}
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1480  Vcvv 2686  {csn 3527  ⟨cop 3530   × cxp 4537  Rel wrel 4544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546 This theorem is referenced by:  cnvsn  5021  fsn  5592
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