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Theorem atpointN 29305
Description: The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
ispoint.a  |-  A  =  ( Atoms `  K )
ispoint.p  |-  P  =  ( Points `  K )
Assertion
Ref Expression
atpointN  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )

Proof of Theorem atpointN
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  { X }  =  { X }
2 sneq 3651 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
32eqeq2d 2294 . . . . 5  |-  ( x  =  X  ->  ( { X }  =  {
x }  <->  { X }  =  { X } ) )
43rspcev 2884 . . . 4  |-  ( ( X  e.  A  /\  { X }  =  { X } )  ->  E. x  e.  A  { X }  =  { x } )
51, 4mpan2 652 . . 3  |-  ( X  e.  A  ->  E. x  e.  A  { X }  =  { x } )
65adantl 452 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  E. x  e.  A  { X }  =  {
x } )
7 ispoint.a . . . 4  |-  A  =  ( Atoms `  K )
8 ispoint.p . . . 4  |-  P  =  ( Points `  K )
97, 8ispointN 29304 . . 3  |-  ( K  e.  D  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
109adantr 451 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
116, 10mpbird 223 1  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {csn 3640   ` cfv 5255   Atomscatm 28826   PointscpointsN 29057
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 29064
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