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Theorem mrissmrcd 14576
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 14563, and so are equal by mrieqv2d 14575.) (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 14563 . . . . 5  |-  ( ph  ->  ( N `  S
)  =  ( N `
 T ) )
6 pssne 3450 . . . . . . 7  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  T )  =/=  ( N `  S
) )
76necomd 2693 . . . . . 6  |-  ( ( N `  T ) 
C.  ( N `  S )  ->  ( N `  S )  =/=  ( N `  T
) )
87necon2bi 2655 . . . . 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 14572 . . . . . . 7  |-  ( ph  ->  S  C_  X )
131, 2, 11, 12mrieqv2d 14575 . . . . . 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 4437 . . . . . 6  |-  ( ph  ->  T  e.  _V )
16 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  s  =  T )
1716psseq1d 3446 . . . . . . 7  |-  ( (
ph  /\  s  =  T )  ->  (
s  C.  S  <->  T  C.  S
) )
1816fveq2d 5693 . . . . . . . 8  |-  ( (
ph  /\  s  =  T )  ->  ( N `  s )  =  ( N `  T ) )
1918psseq1d 3446 . . . . . . 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 3053 . . . . 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 3453 . . . . 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 2446 1  |-  ( ph  ->  S  =  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   _Vcvv 2970    C_ wss 3326    C. wpss 3327   ` cfv 5416  Moorecmre 14518  mrClscmrc 14519  mrIndcmri 14520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-int 4127  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-mre 14522  df-mrc 14523  df-mri 14524
This theorem is referenced by:  mreexexlem3d  14582  acsmap2d  15347
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