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Theorem 0psubN 30720
Description: The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
0psub.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
0psubN  |-  ( K  e.  V  ->  (/)  e.  S
)

Proof of Theorem 0psubN
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ss 3644 . . 3  |-  (/)  C_  ( Atoms `  K )
2 ral0 3760 . . 3  |-  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  (
Atoms `  K ) ( r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) )
31, 2pm3.2i 443 . 2  |-  ( (/)  C_  ( Atoms `  K )  /\  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  ( Atoms `  K )
( r ( le
`  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) ) )
4 eqid 2443 . . 3  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2443 . . 3  |-  ( join `  K )  =  (
join `  K )
6 eqid 2443 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
7 0psub.s . . 3  |-  S  =  ( PSubSp `  K )
84, 5, 6, 7ispsubsp 30716 . 2  |-  ( K  e.  V  ->  ( (/) 
e.  S  <->  ( (/)  C_  ( Atoms `  K )  /\  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  ( Atoms `  K )
( r ( le
`  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) ) ) ) )
93, 8mpbiri 226 1  |-  ( K  e.  V  ->  (/)  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   A.wral 2712    C_ wss 3309   (/)c0 3616   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   lecple 13574   joincjn 14439   Atomscatm 30235   PSubSpcpsubsp 30467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5453  df-fun 5491  df-fv 5497  df-ov 6120  df-psubsp 30474
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