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Theorem 0psubN 30277
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 3643 . . 3  |-  (/)  C_  ( Atoms `  K )
2 ral0 3719 . . 3  |-  A. p  e.  (/)  A. q  e.  (/)  A. r  e.  (
Atoms `  K ) ( r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  (/) )
31, 2pm3.2i 442 . 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 2430 . . 3  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2430 . . 3  |-  ( join `  K )  =  (
join `  K )
6 eqid 2430 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
7 0psub.s . . 3  |-  S  =  ( PSubSp `  K )
84, 5, 6, 7ispsubsp 30273 . 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 225 1  |-  ( K  e.  V  ->  (/)  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2692    C_ wss 3307   (/)c0 3615   class class class wbr 4199   ` cfv 5440  (class class class)co 6067   lecple 13519   joincjn 14384   Atomscatm 29792   PSubSpcpsubsp 30024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-iota 5404  df-fun 5442  df-fv 5448  df-ov 6070  df-psubsp 30031
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