Theorem mrieqvd 16298

Description: In a Moore system, a set is independent if and only if, for all elements of the set, the closure of the set with the element removed is unequal to the closure of the original set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)

Hypotheses

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

Assertion

Ref	Expression
mrieqvd	|- ( ph -> ( S e. I <-> A. x e. S ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )

Distinct variable groups:   x, A   x, S   ph, x

Allowed substitution hints:   I( x)   N( x)   X( x)

Proof of Theorem mrieqvd

Step	Hyp	Ref	Expression
1		mrieqvd.2	. . 3 |- N = (mrCls ` A )
2		mrieqvd.3	. . 3 |- I = (mrInd ` A )
3		mrieqvd.1	. . 3 |- ( ph -> A e. (Moore ` X ) )
4		mrieqvd.4	. . 3 |- ( ph -> S C_ X )
5	1, 2, 3, 4	ismri2d 16293	. 2 |- ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
6	3	adantr 481	. . . . 5 |- ( ( ph /\ x e. S ) -> A e. (Moore ` X ) )
7	4	adantr 481	. . . . 5 |- ( ( ph /\ x e. S ) -> S C_ X )
8		simpr 477	. . . . 5 |- ( ( ph /\ x e. S ) -> x e. S )
9	6, 1, 7, 8	mrieqvlemd 16289	. . . 4 |- ( ( ph /\ x e. S ) -> ( x e. ( N ` ( S \ { x } ) ) <-> ( N ` ( S \ { x } ) ) = ( N ` S ) ) )
10	9	necon3bbid 2831	. . 3 |- ( ( ph /\ x e. S ) -> ( -. x e. ( N ` ( S \ { x } ) ) <-> ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )
11	10	ralbidva 2985	. 2 |- ( ph -> ( A. x e. S -. x e. ( N ` ( S \ { x } ) ) <-> A. x e. S ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )
12	5, 11	bitrd 268	1 |- ( ph -> ( S e. I <-> A. x e. S ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   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: (None)