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Theorem mrissmrcd 13793
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 13780, and so are equal by mrieqv2d 13792.) (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 13780 . . . . 5  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
6 pssne 3387 . . . . . . 7  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  T )  =/=  ( N `  S
) )
76necomd 2634 . . . . . 6  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  S )  =/=  ( N `  T
) )
87necon2bi 2597 . . . . 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 13789 . . . . . . 7  |-  ( ph  ->  S  C_  X )
131, 2, 11, 12mrieqv2d 13792 . . . . . 6  |-  ( ph  ->  ( S  e.  I  <->  A. s ( s  C.  S  ->  ( N `  s )  C.  ( N `  S )
) ) )
1410, 13mpbid 202 . . . . 5  |-  ( ph  ->  A. s ( s 
C.  S  ->  ( N `  s )  C.  ( N `  S
) ) )
1510, 4ssexd 4292 . . . . . 6  |-  ( ph  ->  T  e.  _V )
16 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  s  =  T )
1716psseq1d 3383 . . . . . . 7  |-  ( (
ph  /\  s  =  T )  ->  (
s  C.  S  <->  T  C.  S ) )
1816fveq2d 5673 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  ( N `  s )  =  ( N `  T ) )
1918psseq1d 3383 . . . . . . 7  |-  ( (
ph  /\  s  =  T )  ->  (
( N `  s
)  C.  ( N `  S )  <->  ( N `  T )  C.  ( N `  S )
) )
2017, 19imbi12d 312 . . . . . 6  |-  ( (
ph  /\  s  =  T )  ->  (
( s  C.  S  ->  ( N `  s
)  C.  ( N `  S ) )  <->  ( T  C.  S  ->  ( N `
 T )  C.  ( N `  S ) ) ) )
2115, 20spcdv 2978 . . . . 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 170 . . 3  |-  ( ph  ->  -.  T  C.  S
)
24 sspss 3390 . . . . 5  |-  ( T 
C_  S  <->  ( T  C.  S  \/  T  =  S ) )
254, 24sylib 189 . . . 4  |-  ( ph  ->  ( T  C.  S  \/  T  =  S
) )
2625ord 367 . . 3  |-  ( ph  ->  ( -.  T  C.  S  ->  T  =  S ) )
2723, 26mpd 15 . 2  |-  ( ph  ->  T  =  S )
2827eqcomd 2393 1  |-  ( ph  ->  S  =  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   _Vcvv 2900    C_ wss 3264    C. wpss 3265   ` cfv 5395  Moorecmre 13735  mrClscmrc 13736  mrIndcmri 13737
This theorem is referenced by:  mreexexlem3d  13799  acsmap2d  14533
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-mre 13739  df-mrc 13740  df-mri 13741
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