Theorem ismri2 16292

Description: Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ismri2.1	|- N = (mrCls ` A )
ismri2.2	|- I = (mrInd ` A )

Assertion

Ref	Expression
ismri2	|- ( ( A e. (Moore ` X ) /\ S C_ X ) -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )

Distinct variable groups:   x, A   x, S

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

Proof of Theorem ismri2

Step	Hyp	Ref	Expression
1		ismri2.1	. . 3 |- N = (mrCls ` A )
2		ismri2.2	. . 3 |- I = (mrInd ` A )
3	1, 2	ismri 16291	. 2 |- ( A e. (Moore ` X ) -> ( S e. I <-> ( S C_ X /\ A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) ) )
4	3	baibd 948	1 |- ( ( A e. (Moore ` X ) /\ S C_ X ) -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = 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-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-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-mre 16246  df-mri 16248

This theorem is referenced by:  ismri2d  16293  lindsdom  33403  aacllem  42547