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Theorem atpointN 30602
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 2438 . . . 4  |-  { X }  =  { X }
2 sneq 3827 . . . . . 6  |-  ( x  =  X  ->  { x }  =  { X } )
32eqeq2d 2449 . . . . 5  |-  ( x  =  X  ->  ( { X }  =  {
x }  <->  { X }  =  { X } ) )
43rspcev 3054 . . . 4  |-  ( ( X  e.  A  /\  { X }  =  { X } )  ->  E. x  e.  A  { X }  =  { x } )
51, 4mpan2 654 . . 3  |-  ( X  e.  A  ->  E. x  e.  A  { X }  =  { x } )
65adantl 454 . 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 30601 . . 3  |-  ( K  e.  D  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
109adantr 453 . 2  |-  ( ( K  e.  D  /\  X  e.  A )  ->  ( { X }  e.  P  <->  E. x  e.  A  { X }  =  {
x } ) )
116, 10mpbird 225 1  |-  ( ( K  e.  D  /\  X  e.  A )  ->  { X }  e.  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   {csn 3816   ` cfv 5456   Atomscatm 30123   PointscpointsN 30354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-pointsN 30361
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