Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ispointN Unicode version

Theorem ispointN 29931
Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a  |-  A  =  ( Atoms `  K )
ispoint.p  |-  P  =  ( Points `  K )
Assertion
Ref Expression
ispointN  |-  ( K  e.  D  ->  ( X  e.  P  <->  E. a  e.  A  X  =  { a } ) )
Distinct variable groups:    A, a    X, a
Allowed substitution hints:    D( a)    P( a)    K( a)

Proof of Theorem ispointN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ispoint.a . . . 4  |-  A  =  ( Atoms `  K )
2 ispoint.p . . . 4  |-  P  =  ( Points `  K )
31, 2pointsetN 29930 . . 3  |-  ( K  e.  D  ->  P  =  { x  |  E. a  e.  A  x  =  { a } }
)
43eleq2d 2350 . 2  |-  ( K  e.  D  ->  ( X  e.  P  <->  X  e.  { x  |  E. a  e.  A  x  =  { a } }
) )
5 snex 4216 . . . . 5  |-  { a }  e.  _V
6 eleq1 2343 . . . . 5  |-  ( X  =  { a }  ->  ( X  e. 
_V 
<->  { a }  e.  _V ) )
75, 6mpbiri 224 . . . 4  |-  ( X  =  { a }  ->  X  e.  _V )
87rexlimivw 2663 . . 3  |-  ( E. a  e.  A  X  =  { a }  ->  X  e.  _V )
9 eqeq1 2289 . . . 4  |-  ( x  =  X  ->  (
x  =  { a }  <->  X  =  {
a } ) )
109rexbidv 2564 . . 3  |-  ( x  =  X  ->  ( E. a  e.  A  x  =  { a } 
<->  E. a  e.  A  X  =  { a } ) )
118, 10elab3 2921 . 2  |-  ( X  e.  { x  |  E. a  e.  A  x  =  { a } }  <->  E. a  e.  A  X  =  { a } )
124, 11syl6bb 252 1  |-  ( K  e.  D  ->  ( X  e.  P  <->  E. a  e.  A  X  =  { a } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   {csn 3640   ` cfv 5255   Atomscatm 29453   PointscpointsN 29684
This theorem is referenced by:  atpointN  29932  pointpsubN  29940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-pointsN 29691
  Copyright terms: Public domain W3C validator