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Theorem psubclssatN 30423
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 2404 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
3 psubclssat.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3psubcliN 30420 . 2  |-  ( ( K  e.  D  /\  X  e.  C )  ->  ( X  C_  A  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) )
54simpld 446 1  |-  ( ( K  e.  D  /\  X  e.  C )  ->  X  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280   ` cfv 5413   Atomscatm 29746   _|_ PcpolN 30384   PSubClcpscN 30416
This theorem is referenced by:  pmapidclN  30424  psubclinN  30430  paddatclN  30431  pclfinclN  30432  poml6N  30437  osumcllem3N  30440  osumcllem9N  30446  osumcllem11N  30448  osumclN  30449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-psubclN 30417
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