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Theorem mreexdomd 13567
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 13566. (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 13554 . . . 4  |-  ( ph  ->  S  C_  X )
7 dif0 3537 . . . 4  |-  ( X 
\  (/) )  =  X
86, 7syl6sseqr 3238 . . 3  |-  ( ph  ->  S  C_  ( X  \  (/) ) )
9 mreexdomd.6 . . . 4  |-  ( ph  ->  T  C_  X )
109, 7syl6sseqr 3238 . . 3  |-  ( ph  ->  T  C_  ( X  \  (/) ) )
11 mreexdomd.5 . . . 4  |-  ( ph  ->  S  C_  ( N `  T ) )
12 un0 3492 . . . . 5  |-  ( T  u.  (/) )  =  T
1312fveq2i 5544 . . . 4  |-  ( N `
 ( T  u.  (/) ) )  =  ( N `  T )
1411, 13syl6sseqr 3238 . . 3  |-  ( ph  ->  S  C_  ( N `  ( T  u.  (/) ) ) )
15 un0 3492 . . . 4  |-  ( S  u.  (/) )  =  S
1615, 5syl5eqel 2380 . . 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 13566 . 2  |-  ( ph  ->  E. i  e.  ~P  T ( S  ~~  i  /\  ( i  u.  (/) )  e.  I
) )
19 simprrl 740 . . 3  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~~  i
)
20 simprl 732 . . . . 5  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  e.  ~P T )
2120elpwid 3647 . . . 4  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  C_  T
)
221elfvexd 5572 . . . . . . 7  |-  ( ph  ->  X  e.  _V )
2322, 9ssexd 4177 . . . . . 6  |-  ( ph  ->  T  e.  _V )
24 ssdomg 6923 . . . . . 6  |-  ( T  e.  _V  ->  (
i  C_  T  ->  i  ~<_  T ) )
2523, 24syl 15 . . . . 5  |-  ( ph  ->  ( i  C_  T  ->  i  ~<_  T ) )
2625adantr 451 . . . 4  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  ( i  C_  T  ->  i  ~<_  T ) )
2721, 26mpd 14 . . 3  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  ~<_  T )
28 endomtr 6935 . . 3  |-  ( ( S  ~~  i  /\  i  ~<_  T )  ->  S  ~<_  T )
2919, 27, 28syl2anc 642 . 2  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~<_  T )
3018, 29rexlimddv 2684 1  |-  ( ph  ->  S  ~<_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039   ` cfv 5271    ~~ cen 6876    ~<_ cdom 6877   Fincfn 6879  Moorecmre 13500  mrClscmrc 13501  mrIndcmri 13502
This theorem is referenced by:  mreexfidimd  13568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-ac2 8105
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 6320  df-recs 6404  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-ac 7759  df-mre 13504  df-mrc 13505  df-mri 13506
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