ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  relsnop GIF version

Theorem relsnop 4729
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 4673 . 2 𝐴, 𝐵⟩ ∈ (V × V)
41, 2opex 4226 . . 3 𝐴, 𝐵⟩ ∈ V
54relsn 4728 . 2 (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V))
63, 5mpbir 146 1 Rel {⟨𝐴, 𝐵⟩}
Colors of variables: wff set class
Syntax hints:  wcel 2148  Vcvv 2737  {csn 3591  cop 3594   × cxp 4621  Rel wrel 4628
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-xp 4629  df-rel 4630
This theorem is referenced by:  cnvsn  5107  fsn  5684
  Copyright terms: Public domain W3C validator