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

Theorem relsn 5364
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 5362 . 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 196  wcel 2145  Vcvv 3351  {csn 4317   × cxp 5248  Rel wrel 5255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737  df-sn 4318  df-rel 5257
This theorem is referenced by:  relsnopOLD  5367  relsn2OLD  5746  setscom  16110  setsid  16121
  Copyright terms: Public domain W3C validator