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Theorem psubclssatN 30199
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclssat.a  |-  A  =  ( Atoms `  K )
psubclssat.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
psubclssatN  |-  ( ( K  e.  D  /\  X  e.  C )  ->  X  C_  A )

Proof of Theorem psubclssatN
StepHypRef Expression
1 psubclssat.a . . 3  |-  A  =  ( Atoms `  K )
2 eqid 2358 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
3 psubclssat.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3psubcliN 30196 . 2  |-  ( ( K  e.  D  /\  X  e.  C )  ->  ( X  C_  A  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) )
54simpld 445 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  X  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228   ` cfv 5337   Atomscatm 29522   _|_ PcpolN 30160   PSubClcpscN 30192
This theorem is referenced by:  pmapidclN  30200  psubclinN  30206  paddatclN  30207  pclfinclN  30208  poml6N  30213  osumcllem3N  30216  osumcllem9N  30222  osumcllem11N  30224  osumclN  30225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-psubclN 30193
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