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Theorem mreexdomd 15263
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 15262. (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 15250 . . . 4  |-  ( ph  ->  S  C_  X )
7 dif0 3842 . . . 4  |-  ( X 
\  (/) )  =  X
86, 7syl6sseqr 3489 . . 3  |-  ( ph  ->  S  C_  ( X  \  (/) ) )
9 mreexdomd.6 . . . 4  |-  ( ph  ->  T  C_  X )
109, 7syl6sseqr 3489 . . 3  |-  ( ph  ->  T  C_  ( X  \  (/) ) )
11 mreexdomd.5 . . . 4  |-  ( ph  ->  S  C_  ( N `  T ) )
12 un0 3764 . . . . 5  |-  ( T  u.  (/) )  =  T
1312fveq2i 5852 . . . 4  |-  ( N `
 ( T  u.  (/) ) )  =  ( N `  T )
1411, 13syl6sseqr 3489 . . 3  |-  ( ph  ->  S  C_  ( N `  ( T  u.  (/) ) ) )
15 un0 3764 . . . 4  |-  ( S  u.  (/) )  =  S
1615, 5syl5eqel 2494 . . 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 15262 . 2  |-  ( ph  ->  E. i  e.  ~P  T ( S  ~~  i  /\  ( i  u.  (/) )  e.  I
) )
19 simprrl 766 . . 3  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~~  i
)
20 simprl 756 . . . . 5  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  e.  ~P T )
2120elpwid 3965 . . . 4  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  C_  T
)
221elfvexd 5877 . . . . . . 7  |-  ( ph  ->  X  e.  _V )
2322, 9ssexd 4541 . . . . . 6  |-  ( ph  ->  T  e.  _V )
24 ssdomg 7599 . . . . . 6  |-  ( T  e.  _V  ->  (
i  C_  T  ->  i  ~<_  T ) )
2523, 24syl 17 . . . . 5  |-  ( ph  ->  ( i  C_  T  ->  i  ~<_  T ) )
2625adantr 463 . . . 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 7611 . . 3  |-  ( ( S  ~~  i  /\  i  ~<_  T )  ->  S  ~<_  T )
2919, 27, 28syl2anc 659 . 2  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  S  ~<_  T )
3018, 29rexlimddv 2900 1  |-  ( ph  ->  S  ~<_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   _Vcvv 3059    \ cdif 3411    u. cun 3412    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   {csn 3972   class class class wbr 4395   ` cfv 5569    ~~ cen 7551    ~<_ cdom 7552   Fincfn 7554  Moorecmre 15196  mrClscmrc 15197  mrIndcmri 15198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-ac2 8875
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-om 6684  df-wrecs 7013  df-recs 7075  df-1o 7167  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-ac 8529  df-mre 15200  df-mrc 15201  df-mri 15202
This theorem is referenced by:  mreexfidimd  15264
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