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Theorem mrissmrcd 15056
Description: In a Moore system, if an independent set is between a set and its closure, the two sets are equal (since the two sets must have equal closures by mressmrcd 15043, and so are equal by mrieqv2d 15055.) (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrissmrcd.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mrissmrcd.2  |-  N  =  (mrCls `  A )
mrissmrcd.3  |-  I  =  (mrInd `  A )
mrissmrcd.4  |-  ( ph  ->  S  C_  ( N `  T ) )
mrissmrcd.5  |-  ( ph  ->  T  C_  S )
mrissmrcd.6  |-  ( ph  ->  S  e.  I )
Assertion
Ref Expression
mrissmrcd  |-  ( ph  ->  S  =  T )

Proof of Theorem mrissmrcd
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 mrissmrcd.1 . . . . . 6  |-  ( ph  ->  A  e.  (Moore `  X ) )
2 mrissmrcd.2 . . . . . 6  |-  N  =  (mrCls `  A )
3 mrissmrcd.4 . . . . . 6  |-  ( ph  ->  S  C_  ( N `  T ) )
4 mrissmrcd.5 . . . . . 6  |-  ( ph  ->  T  C_  S )
51, 2, 3, 4mressmrcd 15043 . . . . 5  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
6 pssne 3596 . . . . . . 7  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  T )  =/=  ( N `  S
) )
76necomd 2728 . . . . . 6  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  S )  =/=  ( N `  T
) )
87necon2bi 2694 . . . . 5  |-  ( ( N `  S )  =  ( N `  T )  ->  -.  ( N `  T ) 
C.  ( N `  S ) )
95, 8syl 16 . . . 4  |-  ( ph  ->  -.  ( N `  T )  C.  ( N `  S )
)
10 mrissmrcd.6 . . . . . 6  |-  ( ph  ->  S  e.  I )
11 mrissmrcd.3 . . . . . . 7  |-  I  =  (mrInd `  A )
1211, 1, 10mrissd 15052 . . . . . . 7  |-  ( ph  ->  S  C_  X )
131, 2, 11, 12mrieqv2d 15055 . . . . . 6  |-  ( ph  ->  ( S  e.  I  <->  A. s ( s  C.  S  ->  ( N `  s )  C.  ( N `  S )
) ) )
1410, 13mpbid 210 . . . . 5  |-  ( ph  ->  A. s ( s 
C.  S  ->  ( N `  s )  C.  ( N `  S
) ) )
1510, 4ssexd 4603 . . . . . 6  |-  ( ph  ->  T  e.  _V )
16 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  s  =  T )
1716psseq1d 3592 . . . . . . 7  |-  ( (
ph  /\  s  =  T )  ->  (
s  C.  S  <->  T  C.  S
) )
1816fveq2d 5876 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  ( N `  s )  =  ( N `  T ) )
1918psseq1d 3592 . . . . . . 7  |-  ( (
ph  /\  s  =  T )  ->  (
( N `  s
)  C.  ( N `  S )  <->  ( N `  T )  C.  ( N `  S )
) )
2017, 19imbi12d 320 . . . . . 6  |-  ( (
ph  /\  s  =  T )  ->  (
( s  C.  S  ->  ( N `  s
)  C.  ( N `  S ) )  <->  ( T  C.  S  ->  ( N `  T )  C.  ( N `  S )
) ) )
2115, 20spcdv 3192 . . . . 5  |-  ( ph  ->  ( A. s ( s  C.  S  ->  ( N `  s ) 
C.  ( N `  S ) )  -> 
( T  C.  S  ->  ( N `  T
)  C.  ( N `  S ) ) ) )
2214, 21mpd 15 . . . 4  |-  ( ph  ->  ( T  C.  S  ->  ( N `  T
)  C.  ( N `  S ) ) )
239, 22mtod 177 . . 3  |-  ( ph  ->  -.  T  C.  S
)
24 sspss 3599 . . . . 5  |-  ( T 
C_  S  <->  ( T  C.  S  \/  T  =  S ) )
254, 24sylib 196 . . . 4  |-  ( ph  ->  ( T  C.  S  \/  T  =  S
) )
2625ord 377 . . 3  |-  ( ph  ->  ( -.  T  C.  S  ->  T  =  S ) )
2723, 26mpd 15 . 2  |-  ( ph  ->  T  =  S )
2827eqcomd 2465 1  |-  ( ph  ->  S  =  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471    C. wpss 3472   ` cfv 5594  Moorecmre 14998  mrClscmrc 14999  mrIndcmri 15000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 15002  df-mrc 15003  df-mri 15004
This theorem is referenced by:  mreexexlem3d  15062  acsmap2d  15935
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