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

Theorem relsn 4872
Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
Hypothesis
Ref Expression
relsn.1  |-  A  e. 
_V
Assertion
Ref Expression
relsn  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)

Proof of Theorem relsn
StepHypRef Expression
1 df-rel 4778 . 2  |-  ( Rel 
{ A }  <->  { A }  C_  ( _V  X.  _V ) )
2 relsn.1 . . 3  |-  A  e. 
_V
32snss 3824 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  { A }  C_  ( _V  X.  _V )
)
41, 3bitr4i 243 1  |-  ( Rel 
{ A }  <->  A  e.  ( _V  X.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1710   _Vcvv 2864    C_ wss 3228   {csn 3716    X. cxp 4769   Rel wrel 4776
This theorem is referenced by:  relsnop  4873  relsn2  5225  setscom  13273  setsid  13284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-v 2866  df-in 3235  df-ss 3242  df-sn 3722  df-rel 4778
  Copyright terms: Public domain W3C validator