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

Theorem ispointN 33725
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 33724 . . 3  |-  ( K  e.  D  ->  P  =  { x  |  E. a  e.  A  x  =  { a } }
)
43eleq2d 2524 . 2  |-  ( K  e.  D  ->  ( X  e.  P  <->  X  e.  { x  |  E. a  e.  A  x  =  { a } }
) )
5 snex 4642 . . . . 5  |-  { a }  e.  _V
6 eleq1 2526 . . . . 5  |-  ( X  =  { a }  ->  ( X  e. 
_V 
<->  { a }  e.  _V ) )
75, 6mpbiri 233 . . . 4  |-  ( X  =  { a }  ->  X  e.  _V )
87rexlimivw 2943 . . 3  |-  ( E. a  e.  A  X  =  { a }  ->  X  e.  _V )
9 eqeq1 2458 . . . 4  |-  ( x  =  X  ->  (
x  =  { a }  <->  X  =  {
a } ) )
109rexbidv 2868 . . 3  |-  ( x  =  X  ->  ( E. a  e.  A  x  =  { a } 
<->  E. a  e.  A  X  =  { a } ) )
118, 10elab3 3220 . 2  |-  ( X  e.  { x  |  E. a  e.  A  x  =  { a } }  <->  E. a  e.  A  X  =  { a } )
124, 11syl6bb 261 1  |-  ( K  e.  D  ->  ( X  e.  P  <->  E. a  e.  A  X  =  { a } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   {cab 2439   E.wrex 2800   _Vcvv 3078   {csn 3986   ` cfv 5527   Atomscatm 33247   PointscpointsN 33478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-pointsN 33485
This theorem is referenced by:  atpointN  33726  pointpsubN  33734
  Copyright terms: Public domain W3C validator