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Theorem mreexdomd 14608
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 14607. (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 14595 . . . 4  |-  ( ph  ->  S  C_  X )
7 dif0 3770 . . . 4  |-  ( X 
\  (/) )  =  X
86, 7syl6sseqr 3424 . . 3  |-  ( ph  ->  S  C_  ( X  \  (/) ) )
9 mreexdomd.6 . . . 4  |-  ( ph  ->  T  C_  X )
109, 7syl6sseqr 3424 . . 3  |-  ( ph  ->  T  C_  ( X  \  (/) ) )
11 mreexdomd.5 . . . 4  |-  ( ph  ->  S  C_  ( N `  T ) )
12 un0 3683 . . . . 5  |-  ( T  u.  (/) )  =  T
1312fveq2i 5715 . . . 4  |-  ( N `
 ( T  u.  (/) ) )  =  ( N `  T )
1411, 13syl6sseqr 3424 . . 3  |-  ( ph  ->  S  C_  ( N `  ( T  u.  (/) ) ) )
15 un0 3683 . . . 4  |-  ( S  u.  (/) )  =  S
1615, 5syl5eqel 2527 . . 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 14607 . 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 3891 . . . 4  |-  ( (
ph  /\  ( i  e.  ~P T  /\  ( S  ~~  i  /\  (
i  u.  (/) )  e.  I ) ) )  ->  i  C_  T
)
221elfvexd 5739 . . . . . . 7  |-  ( ph  ->  X  e.  _V )
2322, 9ssexd 4460 . . . . . 6  |-  ( ph  ->  T  e.  _V )
24 ssdomg 7376 . . . . . 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 7388 . . 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 2866 1  |-  ( ph  ->  S  ~<_  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   _Vcvv 2993    \ cdif 3346    u. cun 3347    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   {csn 3898   class class class wbr 4313   ` cfv 5439    ~~ cen 7328    ~<_ cdom 7329   Fincfn 7331  Moorecmre 14541  mrClscmrc 14542  mrIndcmri 14543
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 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-ac2 8653
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-om 6498  df-recs 6853  df-1o 6941  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-card 8130  df-ac 8307  df-mre 14545  df-mrc 14546  df-mri 14547
This theorem is referenced by:  mreexfidimd  14609
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