Theorem mrissmrid 16301

Description: In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
mrissmrid.1	|- ( ph -> A e. (Moore ` X ) )
mrissmrid.2	|- N = (mrCls ` A )
mrissmrid.3	|- I = (mrInd ` A )
mrissmrid.4	|- ( ph -> S e. I )
mrissmrid.5	|- ( ph -> T C_ S )

Assertion

Ref	Expression
mrissmrid	|- ( ph -> T e. I )

Proof of Theorem mrissmrid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrissmrid.2	. 2 |- N = (mrCls ` A )
2		mrissmrid.3	. 2 |- I = (mrInd ` A )
3		mrissmrid.1	. 2 |- ( ph -> A e. (Moore ` X ) )
4		mrissmrid.5	. . 3 |- ( ph -> T C_ S )
5		mrissmrid.4	. . . 4 |- ( ph -> S e. I )
6	2, 3, 5	mrissd 16296	. . 3 |- ( ph -> S C_ X )
7	4, 6	sstrd 3613	. 2 |- ( ph -> T C_ X )
8	1, 2, 3, 6	ismri2d 16293	. . . 4 |- ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
9	5, 8	mpbid 222	. . 3 |- ( ph -> A. x e. S -. x e. ( N ` ( S \ { x } ) ) )
10	4	sseld 3602	. . . . 5 |- ( ph -> ( x e. T -> x e. S ) )
11	4	ssdifd 3746	. . . . . . 7 |- ( ph -> ( T \ { x } ) C_ ( S \ { x } ) )
12	6	ssdifssd 3748	. . . . . . 7 |- ( ph -> ( S \ { x } ) C_ X )
13	3, 1, 11, 12	mrcssd 16284	. . . . . 6 |- ( ph -> ( N ` ( T \ { x } ) ) C_ ( N ` ( S \ { x } ) ) )
14	13	ssneld 3605	. . . . 5 |- ( ph -> ( -. x e. ( N ` ( S \ { x } ) ) -> -. x e. ( N ` ( T \ { x } ) ) ) )
15	10, 14	imim12d 81	. . . 4 |- ( ph -> ( ( x e. S -> -. x e. ( N ` ( S \ { x } ) ) ) -> ( x e. T -> -. x e. ( N ` ( T \ { x } ) ) ) ) )
16	15	ralimdv2 2961	. . 3 |- ( ph -> ( A. x e. S -. x e. ( N ` ( S \ { x } ) ) -> A. x e. T -. x e. ( N ` ( T \ { x } ) ) ) )
17	9, 16	mpd 15	. 2 |- ( ph -> A. x e. T -. x e. ( N ` ( T \ { x } ) ) )
18	1, 2, 3, 7, 17	ismri2dd 16294	1 |- ( ph -> T e. I )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177   ` cfv 5888  Moorecmre 16242  mrClscmrc 16243  mrIndcmri 16244

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-8 1992  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  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-int 4476  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-mre 16246  df-mrc 16247  df-mri 16248

This theorem is referenced by:  mreexexlem2d  16305  acsfiindd  17177