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Theorem 0psubN 28627
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
StepHypRef Expression
1 0ss 3390 . . 3  |-  (/)  C_  ( Atoms `  K )
2 ral0 3464 . . 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 2253 . . 3  |-  ( le
`  K )  =  ( le `  K
)
5 eqid 2253 . . 3  |-  ( join `  K )  =  (
join `  K )
6 eqid 2253 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
7 0psub.s . . 3  |-  S  =  ( PSubSp `  K )
84, 5, 6, 7ispsubsp 28623 . 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2509    C_ wss 3078   (/)c0 3362   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   lecple 13089   joincjn 13922   Atomscatm 28142   PSubSpcpsubsp 28374
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-psubsp 28381
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