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Theorem psubatN 30566
Description: A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
atpsub.a  |-  A  =  ( Atoms `  K )
atpsub.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
psubatN  |-  ( ( K  e.  B  /\  X  e.  S  /\  Y  e.  X )  ->  Y  e.  A )

Proof of Theorem psubatN
StepHypRef Expression
1 atpsub.a . . . 4  |-  A  =  ( Atoms `  K )
2 atpsub.s . . . 4  |-  S  =  ( PSubSp `  K )
31, 2psubssat 30565 . . 3  |-  ( ( K  e.  B  /\  X  e.  S )  ->  X  C_  A )
43sseld 3192 . 2  |-  ( ( K  e.  B  /\  X  e.  S )  ->  ( Y  e.  X  ->  Y  e.  A ) )
543impia 1148 1  |-  ( ( K  e.  B  /\  X  e.  S  /\  Y  e.  X )  ->  Y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271   Atomscatm 30075   PSubSpcpsubsp 30307
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-psubsp 30314
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