Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  relsn Structured version   Visualization version   GIF version

Theorem relsn 5645
 Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
Hypothesis
Ref Expression
relsn.1 𝐴 ∈ V
Assertion
Ref Expression
relsn (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Proof of Theorem relsn
StepHypRef Expression
1 relsn.1 . 2 𝐴 ∈ V
2 relsng 5642 . 2 (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
31, 2ax-mp 5 1 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2112  Vcvv 3444  {csn 4528   × cxp 5521  Rel wrel 5528 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-sn 4529  df-rel 5530 This theorem is referenced by:  setscom  16522  setsid  16533
 Copyright terms: Public domain W3C validator