Theorem 0psubN 35035

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.

Step	Hyp	Ref	Expression
1		0ss 3972	. . 3 |- (/) C_ ( Atoms ` K )
2		ral0 4076	. . 3 |- A. p e. (/) A. q e. (/) A. r e. ( Atoms ` K ) ( r ( le ` K ) ( p ( join ` K ) q ) -> r e. (/) )
3	1, 2	pm3.2i 471	. 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 2622	. . 3 |- ( le ` K ) = ( le ` K )
5		eqid 2622	. . 3 |- ( join ` K ) = ( join ` K )
6		eqid 2622	. . 3 |- ( Atoms ` K ) = ( Atoms ` K )
7		0psub.s	. . 3 |- S = ( PSubSp ` K )
8	4, 5, 6, 7	ispsubsp 35031	. 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. (/) ) ) ) )
9	3, 8	mpbiri 248	1 |- ( K e. V -> (/) e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   lecple 15948   joincjn 16944   Atomscatm 34550   PSubSpcpsubsp 34782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-psubsp 34789

This theorem is referenced by: (None)