Theorem mrieqvd 13818

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 13813	. 2 |- ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
6	3	adantr 452	. . . . 5 |- ( ( ph /\ x e. S ) -> A e. (Moore ` X ) )
7	4	adantr 452	. . . . 5 |- ( ( ph /\ x e. S ) -> S C_ X )
8		simpr 448	. . . . 5 |- ( ( ph /\ x e. S ) -> x e. S )
9	6, 1, 7, 8	mrieqvlemd 13809	. . . 4 |- ( ( ph /\ x e. S ) -> ( x e. ( N ` ( S \ { x } ) ) <-> ( N ` ( S \ { x } ) ) = ( N ` S ) ) )
10	9	necon3bbid 2601	. . 3 |- ( ( ph /\ x e. S ) -> ( -. x e. ( N ` ( S \ { x } ) ) <-> ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )
11	10	ralbidva 2682	. 2 |- ( ph -> ( A. x e. S -. x e. ( N ` ( S \ { x } ) ) <-> A. x e. S ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )
12	5, 11	bitrd 245	1 |- ( ph -> ( S e. I <-> A. x e. S ( N ` ( S \ { x } ) ) =/= ( N ` S ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   \ cdif 3277   C_ wss 3280   {csn 3774   ` cfv 5413  Moorecmre 13762  mrClscmrc 13763  mrIndcmri 13764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-mre 13766  df-mrc 13767  df-mri 13768