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Metamath Proof Explorer

Theorem ismri2dad

Description: Consequence of a set in a Moore system being independent. Deduction form. (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypotheses ismri2dad.1 
|- N = ( mrCls ` A )
ismri2dad.2 
|- I = ( mrInd ` A )
ismri2dad.3 
|- ( ph -> A e. ( Moore ` X ) )
ismri2dad.4 
|- ( ph -> S e. I )
ismri2dad.5 
|- ( ph -> Y e. S )
Assertion ismri2dad 
|- ( ph -> -. Y e. ( N ` ( S \ { Y } ) ) )

Proof

Step Hyp Ref Expression
1 ismri2dad.1 
 |-  N = ( mrCls ` A )
2 ismri2dad.2 
 |-  I = ( mrInd ` A )
3 ismri2dad.3 
 |-  ( ph -> A e. ( Moore ` X ) )
4 ismri2dad.4 
 |-  ( ph -> S e. I )
5 ismri2dad.5 
 |-  ( ph -> Y e. S )
6 2 3 4 mrissd 
 |-  ( ph -> S C_ X )
7 1 2 3 6 ismri2d 
 |-  ( ph -> ( S e. I <-> A. x e. S -. x e. ( N ` ( S \ { x } ) ) ) )
8 4 7 mpbid 
 |-  ( ph -> A. x e. S -. x e. ( N ` ( S \ { x } ) ) )
9 simpr 
 |-  ( ( ph /\ x = Y ) -> x = Y )
10 9 sneqd 
 |-  ( ( ph /\ x = Y ) -> { x } = { Y } )
11 10 difeq2d 
 |-  ( ( ph /\ x = Y ) -> ( S \ { x } ) = ( S \ { Y } ) )
12 11 fveq2d 
 |-  ( ( ph /\ x = Y ) -> ( N ` ( S \ { x } ) ) = ( N ` ( S \ { Y } ) ) )
13 9 12 eleq12d 
 |-  ( ( ph /\ x = Y ) -> ( x e. ( N ` ( S \ { x } ) ) <-> Y e. ( N ` ( S \ { Y } ) ) ) )
14 13 notbid 
 |-  ( ( ph /\ x = Y ) -> ( -. x e. ( N ` ( S \ { x } ) ) <-> -. Y e. ( N ` ( S \ { Y } ) ) ) )
15 5 14 rspcdv 
 |-  ( ph -> ( A. x e. S -. x e. ( N ` ( S \ { x } ) ) -> -. Y e. ( N ` ( S \ { Y } ) ) ) )
16 8 15 mpd 
 |-  ( ph -> -. Y e. ( N ` ( S \ { Y } ) ) )