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Theorem ispointN 30553
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 30552 . . 3  |-  ( K  e.  D  ->  P  =  { x  |  E. a  e.  A  x  =  { a } }
)
43eleq2d 2363 . 2  |-  ( K  e.  D  ->  ( X  e.  P  <->  X  e.  { x  |  E. a  e.  A  x  =  { a } }
) )
5 snex 4232 . . . . 5  |-  { a }  e.  _V
6 eleq1 2356 . . . . 5  |-  ( X  =  { a }  ->  ( X  e. 
_V 
<->  { a }  e.  _V ) )
75, 6mpbiri 224 . . . 4  |-  ( X  =  { a }  ->  X  e.  _V )
87rexlimivw 2676 . . 3  |-  ( E. a  e.  A  X  =  { a }  ->  X  e.  _V )
9 eqeq1 2302 . . . 4  |-  ( x  =  X  ->  (
x  =  { a }  <->  X  =  {
a } ) )
109rexbidv 2577 . . 3  |-  ( x  =  X  ->  ( E. a  e.  A  x  =  { a } 
<->  E. a  e.  A  X  =  { a } ) )
118, 10elab3 2934 . 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   {csn 3653   ` cfv 5271   Atomscatm 30075   PointscpointsN 30306
This theorem is referenced by:  atpointN  30554  pointpsubN  30562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-pointsN 30313
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