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Theorem mreexexlem3d 13548
Description: Base case of the induction in mreexexd 13550. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mreexexlem2d.1  |-  ( ph  ->  A  e.  (Moore `  X ) )
mreexexlem2d.2  |-  N  =  (mrCls `  A )
mreexexlem2d.3  |-  I  =  (mrInd `  A )
mreexexlem2d.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 } ) ) )
mreexexlem2d.5  |-  ( ph  ->  F  C_  ( X  \  H ) )
mreexexlem2d.6  |-  ( ph  ->  G  C_  ( X  \  H ) )
mreexexlem2d.7  |-  ( ph  ->  F  C_  ( N `  ( G  u.  H
) ) )
mreexexlem2d.8  |-  ( ph  ->  ( F  u.  H
)  e.  I )
mreexexlem3d.9  |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )
Assertion
Ref Expression
mreexexlem3d  |-  ( ph  ->  E. i  e.  ~P  G ( F  ~~  i  /\  ( i  u.  H )  e.  I
) )
Distinct variable groups:    i, F    i, G    i, H    i, I
Allowed substitution hints:    ph( y, z, i, s)    A( y, z, i, s)    F( y, z, s)    G( y, z, s)    H( y, z, s)    I( y, z, s)    N( y, z, i, s)    X( y, z, i, s)

Proof of Theorem mreexexlem3d
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( (
ph  /\  F  =  (/) )  ->  F  =  (/) )
2 mreexexlem2d.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  (Moore `  X ) )
32adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  A  e.  (Moore `  X ) )
4 mreexexlem2d.2 . . . . . . . . 9  |-  N  =  (mrCls `  A )
5 mreexexlem2d.3 . . . . . . . . 9  |-  I  =  (mrInd `  A )
6 mreexexlem2d.7 . . . . . . . . . . . 12  |-  ( ph  ->  F  C_  ( N `  ( G  u.  H
) ) )
76adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  ( G  u.  H ) ) )
8 simpr 447 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  G  =  (/) )  ->  G  =  (/) )
98uneq1d 3328 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  (
(/)  u.  H )
)
10 uncom 3319 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  (
(/)  u.  H )
11 un0 3479 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  H
1210, 11eqtr3i 2305 . . . . . . . . . . . . 13  |-  ( (/)  u.  H )  =  H
139, 12syl6eq 2331 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  H )
1413fveq2d 5529 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  ( N `  ( G  u.  H
) )  =  ( N `  H ) )
157, 14sseqtrd 3214 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  H )
)
16 mreexexlem2d.8 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  u.  H
)  e.  I )
1716adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  e.  I
)
185, 3, 17mrissd 13538 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  X
)
1918unssbd 3353 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  X
)
203, 4, 19mrcssidd 13527 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( N `  H )
)
2115, 20unssd 3351 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  ( N `  H )
)
22 ssun2 3339 . . . . . . . . . 10  |-  H  C_  ( F  u.  H
)
2322a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( F  u.  H )
)
243, 4, 5, 21, 23, 17mrissmrcd 13542 . . . . . . . 8  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  =  H )
25 ssequn1 3345 . . . . . . . 8  |-  ( F 
C_  H  <->  ( F  u.  H )  =  H )
2624, 25sylibr 203 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  H
)
27 mreexexlem2d.5 . . . . . . . 8  |-  ( ph  ->  F  C_  ( X  \  H ) )
2827adantr 451 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( X  \  H ) )
2926, 28ssind 3393 . . . . . 6  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( H  i^i  ( X  \  H ) ) )
30 disjdif 3526 . . . . . 6  |-  ( H  i^i  ( X  \  H ) )  =  (/)
3129, 30syl6sseq 3224 . . . . 5  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  (/) )
32 ss0b 3484 . . . . 5  |-  ( F 
C_  (/)  <->  F  =  (/) )
3331, 32sylib 188 . . . 4  |-  ( (
ph  /\  G  =  (/) )  ->  F  =  (/) )
34 mreexexlem3d.9 . . . 4  |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )
351, 33, 34mpjaodan 761 . . 3  |-  ( ph  ->  F  =  (/) )
36 0elpw 4180 . . 3  |-  (/)  e.  ~P G
3735, 36syl6eqel 2371 . 2  |-  ( ph  ->  F  e.  ~P G
)
382elfvexd 5556 . . . 4  |-  ( ph  ->  X  e.  _V )
3927difss2d 3306 . . . 4  |-  ( ph  ->  F  C_  X )
4038, 39ssexd 4161 . . 3  |-  ( ph  ->  F  e.  _V )
41 enrefg 6893 . . 3  |-  ( F  e.  _V  ->  F  ~~  F )
4240, 41syl 15 . 2  |-  ( ph  ->  F  ~~  F )
43 breq2 4027 . . . 4  |-  ( i  =  F  ->  ( F  ~~  i  <->  F  ~~  F ) )
44 uneq1 3322 . . . . 5  |-  ( i  =  F  ->  (
i  u.  H )  =  ( F  u.  H ) )
4544eleq1d 2349 . . . 4  |-  ( i  =  F  ->  (
( i  u.  H
)  e.  I  <->  ( F  u.  H )  e.  I
) )
4643, 45anbi12d 691 . . 3  |-  ( i  =  F  ->  (
( F  ~~  i  /\  ( i  u.  H
)  e.  I )  <-> 
( F  ~~  F  /\  ( F  u.  H
)  e.  I ) ) )
4746rspcev 2884 . 2  |-  ( ( F  e.  ~P G  /\  ( F  ~~  F  /\  ( F  u.  H
)  e.  I ) )  ->  E. i  e.  ~P  G ( F 
~~  i  /\  (
i  u.  H )  e.  I ) )
4837, 42, 16, 47syl12anc 1180 1  |-  ( ph  ->  E. i  e.  ~P  G ( F  ~~  i  /\  ( i  u.  H )  e.  I
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {csn 3640   class class class wbr 4023   ` cfv 5255    ~~ cen 6860  Moorecmre 13484  mrClscmrc 13485  mrIndcmri 13486
This theorem is referenced by:  mreexexlem4d  13549  mreexexd  13550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-en 6864  df-mre 13488  df-mrc 13489  df-mri 13490
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