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Theorem 0psubN 29938
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 3483 . . 3  |-  (/)  C_  ( Atoms `  K )
2 ral0 3558 . . 3  |-  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  (
Atoms `  K ) ( r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) )
31, 2pm3.2i 441 . 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 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
6 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
7 0psub.s . . 3  |-  S  =  ( PSubSp `  K )
84, 5, 6, 7ispsubsp 29934 . 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 224 1  |-  ( K  e.  V  ->  (/)  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   PSubSpcpsubsp 29685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-psubsp 29692
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