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Theorem mreexdomd 13875
Description: In a Moore system whose closure operator has the exchange property, if  S is independent and contained in the closure of  T, and either  S or  T is finite, then  T dominates  S. This is an immediate consequence of mreexexd 13874. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mreexdomd.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mreexdomd.2  |-  N  =  (mrCls `  A )
mreexdomd.3  |-  I  =  (mrInd `  A )
mreexdomd.4  |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  (
( N `  (
s  u.  { y } ) )  \ 
( N `  s
) ) y  e.  ( N `  (
s  u.  { z } ) ) )
mreexdomd.5  |-  ( ph  ->  S  C_  ( N `  T ) )
mreexdomd.6  |-  ( ph  ->  T  C_  X )
mreexdomd.7  |-  ( ph  ->  ( S  e.  Fin  \/  T  e.  Fin )
)
mreexdomd.8  |-  ( ph  ->  S  e.  I )
Assertion
Ref Expression
mreexdomd  |-  ( ph  ->  S  ~<_  T )
Distinct variable groups:    X, s,
y, z    ph, s, y, z    I, s, y, z    N, s, y, z
Allowed substitution hints:    A( y, z, s)    S( y, z, s)    T( y, z, s)

Proof of Theorem mreexdomd
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 mreexdomd.1 . . 3  |-  ( ph  ->  A  e.  (Moore `  X ) )
2 mreexdomd.2 . . 3  |-  N  =  (mrCls `  A )
3 mreexdomd.3 . . 3  |-  I  =  (mrInd `  A )
4 mreexdomd.4 . . 3  |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  (
( N `  (
s  u.  { y } ) )  \ 
( N `  s
) ) y  e.  ( N `  (
s  u.  { z } ) ) )
5 mreexdomd.8 . . . . 5  |-  ( ph  ->  S  e.  I )
63, 1, 5mrissd 13862 . . . 4  |-  ( ph  ->  S  C_  X )
7 dif0 3699 . . . 4  |-  ( X 
\  (/) )  =  X
86, 7syl6sseqr 3396 . . 3  |-  ( ph  ->  S  C_  ( X  \  (/) ) )
9 mreexdomd.6 . . . 4  |-  ( ph  ->  T  C_  X )
109, 7syl6sseqr 3396 . . 3  |-  ( ph  ->  T  C_  ( X  \  (/) ) )
11 mreexdomd.5 . . . 4  |-  ( ph  ->  S  C_  ( N `  T ) )
12 un0 3653 . . . . 5  |-  ( T  u.  (/) )  =  T
1312fveq2i 5732 . . . 4  |-  ( N `
 ( T  u.  (/) ) )  =  ( N `  T )
1411, 13syl6sseqr 3396 . . 3  |-  ( ph  ->  S  C_  ( N `  ( T  u.  (/) ) ) )
15 un0 3653 . . . 4  |-  ( S  u.  (/) )  =  S
1615, 5syl5eqel 2521 . . 3  |-  ( ph  ->  ( S  u.  (/) )  e.  I )
17 mreexdomd.7 . . 3  |-  ( ph  ->  ( S  e.  Fin  \/  T  e.  Fin )
)
181, 2, 3, 4, 8, 10, 14, 16, 17mreexexd 13874 . 2  |-  ( ph  ->  E. i  e.  ~P  T ( S  ~~  i  /\  ( i  u.  (/) )  e.  I
) )
19 simprrl 742 . . 3  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~~  i
)
20 simprl 734 . . . . 5  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  e.  ~P T )
2120elpwid 3809 . . . 4  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  C_  T
)
221elfvexd 5760 . . . . . . 7  |-  ( ph  ->  X  e.  _V )
2322, 9ssexd 4351 . . . . . 6  |-  ( ph  ->  T  e.  _V )
24 ssdomg 7154 . . . . . 6  |-  ( T  e.  _V  ->  (
i  C_  T  ->  i  ~<_  T ) )
2523, 24syl 16 . . . . 5  |-  ( ph  ->  ( i  C_  T  ->  i  ~<_  T ) )
2625adantr 453 . . . 4  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  ( i  C_  T  ->  i  ~<_  T ) )
2721, 26mpd 15 . . 3  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  ~<_  T )
28 endomtr 7166 . . 3  |-  ( ( S  ~~  i  /\  i  ~<_  T )  ->  S  ~<_  T )
2919, 27, 28syl2anc 644 . 2  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~<_  T )
3018, 29rexlimddv 2835 1  |-  ( ph  ->  S  ~<_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   _Vcvv 2957    \ cdif 3318    u. cun 3319    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   {csn 3815   class class class wbr 4213   ` cfv 5455    ~~ cen 7107    ~<_ cdom 7108   Fincfn 7110  Moorecmre 13808  mrClscmrc 13809  mrIndcmri 13810
This theorem is referenced by:  mreexfidimd  13876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-ac2 8344
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-riota 6550  df-recs 6634  df-1o 6725  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-card 7827  df-ac 7998  df-mre 13812  df-mrc 13813  df-mri 13814
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