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

Theorem relsnop 4472
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 4418 . 2 𝐴, 𝐵⟩ ∈ (V × V)
4 opexgOLD 3993 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ∈ V)
51, 2, 4mp2an 410 . . 3 𝐴, 𝐵⟩ ∈ V
65relsn 4471 . 2 (Rel {⟨𝐴, 𝐵⟩} ↔ ⟨𝐴, 𝐵⟩ ∈ (V × V))
73, 6mpbir 138 1 Rel {⟨𝐴, 𝐵⟩}
Colors of variables: wff set class
Syntax hints:  wcel 1409  Vcvv 2574  {csn 3403  cop 3406   × cxp 4371  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380
This theorem is referenced by:  cnvsn  4831  fsn  5363
  Copyright terms: Public domain W3C validator