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Theorem ispointN 35567
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 35566 . . 3  |-  ( K  e.  D  ->  P  =  { x  |  E. a  e.  A  x  =  { a } }
)
43eleq2d 2527 . 2  |-  ( K  e.  D  ->  ( X  e.  P  <->  X  e.  { x  |  E. a  e.  A  x  =  { a } }
) )
5 snex 4697 . . . . 5  |-  { a }  e.  _V
6 eleq1 2529 . . . . 5  |-  ( X  =  { a }  ->  ( X  e. 
_V 
<->  { a }  e.  _V ) )
75, 6mpbiri 233 . . . 4  |-  ( X  =  { a }  ->  X  e.  _V )
87rexlimivw 2946 . . 3  |-  ( E. a  e.  A  X  =  { a }  ->  X  e.  _V )
9 eqeq1 2461 . . . 4  |-  ( x  =  X  ->  (
x  =  { a }  <->  X  =  {
a } ) )
109rexbidv 2968 . . 3  |-  ( x  =  X  ->  ( E. a  e.  A  x  =  { a } 
<->  E. a  e.  A  X  =  { a } ) )
118, 10elab3 3253 . 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808   _Vcvv 3109   {csn 4032   ` cfv 5594   Atomscatm 35089   PointscpointsN 35320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-pointsN 35327
This theorem is referenced by:  atpointN  35568  pointpsubN  35576
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