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Theorem mreexdomd 14893
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 14892. (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 14880 . . . 4  |-  ( ph  ->  S  C_  X )
7 dif0 3890 . . . 4  |-  ( X 
\  (/) )  =  X
86, 7syl6sseqr 3544 . . 3  |-  ( ph  ->  S  C_  ( X  \  (/) ) )
9 mreexdomd.6 . . . 4  |-  ( ph  ->  T  C_  X )
109, 7syl6sseqr 3544 . . 3  |-  ( ph  ->  T  C_  ( X  \  (/) ) )
11 mreexdomd.5 . . . 4  |-  ( ph  ->  S  C_  ( N `  T ) )
12 un0 3803 . . . . 5  |-  ( T  u.  (/) )  =  T
1312fveq2i 5860 . . . 4  |-  ( N `
 ( T  u.  (/) ) )  =  ( N `  T )
1411, 13syl6sseqr 3544 . . 3  |-  ( ph  ->  S  C_  ( N `  ( T  u.  (/) ) ) )
15 un0 3803 . . . 4  |-  ( S  u.  (/) )  =  S
1615, 5syl5eqel 2552 . . 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 14892 . 2  |-  ( ph  ->  E. i  e.  ~P  T ( S  ~~  i  /\  ( i  u.  (/) )  e.  I
) )
19 simprrl 763 . . 3  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~~  i
)
20 simprl 755 . . . . 5  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  e.  ~P T )
2120elpwid 4013 . . . 4  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  C_  T
)
221elfvexd 5885 . . . . . . 7  |-  ( ph  ->  X  e.  _V )
2322, 9ssexd 4587 . . . . . 6  |-  ( ph  ->  T  e.  _V )
24 ssdomg 7551 . . . . . 6  |-  ( T  e.  _V  ->  (
i  C_  T  ->  i  ~<_  T ) )
2523, 24syl 16 . . . . 5  |-  ( ph  ->  ( i  C_  T  ->  i  ~<_  T ) )
2625adantr 465 . . . 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 7563 . . 3  |-  ( ( S  ~~  i  /\  i  ~<_  T )  ->  S  ~<_  T )
2919, 27, 28syl2anc 661 . 2  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~<_  T )
3018, 29rexlimddv 2952 1  |-  ( ph  ->  S  ~<_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106    \ cdif 3466    u. cun 3467    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   class class class wbr 4440   ` cfv 5579    ~~ cen 7503    ~<_ cdom 7504   Fincfn 7506  Moorecmre 14826  mrClscmrc 14827  mrIndcmri 14828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-ac2 8832
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-om 6672  df-recs 7032  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-ac 8486  df-mre 14830  df-mrc 14831  df-mri 14832
This theorem is referenced by:  mreexfidimd  14894
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