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Theorem mrissmrcd 13857
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 13844, and so are equal by mrieqv2d 13856.) (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 13844 . . . . 5  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
6 pssne 3435 . . . . . . 7  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  T )  =/=  ( N `  S
) )
76necomd 2681 . . . . . 6  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  S )  =/=  ( N `  T
) )
87necon2bi 2644 . . . . 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 13853 . . . . . . 7  |-  ( ph  ->  S  C_  X )
131, 2, 11, 12mrieqv2d 13856 . . . . . 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 4342 . . . . . 6  |-  ( ph  ->  T  e.  _V )
16 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  s  =  T )
1716psseq1d 3431 . . . . . . 7  |-  ( (
ph  /\  s  =  T )  ->  (
s  C.  S  <->  T  C.  S ) )
1816fveq2d 5724 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  ( N `  s )  =  ( N `  T ) )
1918psseq1d 3431 . . . . . . 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 3026 . . . . 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 3438 . . . . 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 2440 1  |-  ( ph  ->  S  =  T )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312    C. wpss 3313   ` cfv 5446  Moorecmre 13799  mrClscmrc 13800  mrIndcmri 13801
This theorem is referenced by:  mreexexlem3d  13863  acsmap2d  14597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-mre 13803  df-mrc 13804  df-mri 13805
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