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Theorem ispointN 30476
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 30475 . . 3  |-  ( K  e.  D  ->  P  =  { x  |  E. a  e.  A  x  =  { a } }
)
43eleq2d 2502 . 2  |-  ( K  e.  D  ->  ( X  e.  P  <->  X  e.  { x  |  E. a  e.  A  x  =  { a } }
) )
5 snex 4397 . . . . 5  |-  { a }  e.  _V
6 eleq1 2495 . . . . 5  |-  ( X  =  { a }  ->  ( X  e. 
_V 
<->  { a }  e.  _V ) )
75, 6mpbiri 225 . . . 4  |-  ( X  =  { a }  ->  X  e.  _V )
87rexlimivw 2818 . . 3  |-  ( E. a  e.  A  X  =  { a }  ->  X  e.  _V )
9 eqeq1 2441 . . . 4  |-  ( x  =  X  ->  (
x  =  { a }  <->  X  =  {
a } ) )
109rexbidv 2718 . . 3  |-  ( x  =  X  ->  ( E. a  e.  A  x  =  { a } 
<->  E. a  e.  A  X  =  { a } ) )
118, 10elab3 3081 . 2  |-  ( X  e.  { x  |  E. a  e.  A  x  =  { a } }  <->  E. a  e.  A  X  =  { a } )
124, 11syl6bb 253 1  |-  ( K  e.  D  ->  ( X  e.  P  <->  E. a  e.  A  X  =  { a } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948   {csn 3806   ` cfv 5446   Atomscatm 29998   PointscpointsN 30229
This theorem is referenced by:  atpointN  30477  pointpsubN  30485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-pointsN 30236
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