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Theorem mreexexlem3d 13873
Description: Base case of the induction in mreexexd 13875. (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 449 . . . 4  |-  ( (
ph  /\  F  =  (/) )  ->  F  =  (/) )
2 mreexexlem2d.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  (Moore `  X ) )
32adantr 453 . . . . . . . . 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 453 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  ( G  u.  H ) ) )
8 simpr 449 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  G  =  (/) )  ->  G  =  (/) )
98uneq1d 3502 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  (
(/)  u.  H )
)
10 uncom 3493 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  (
(/)  u.  H )
11 un0 3654 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  H
1210, 11eqtr3i 2460 . . . . . . . . . . . . 13  |-  ( (/)  u.  H )  =  H
139, 12syl6eq 2486 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  H )
1413fveq2d 5734 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  ( N `  ( G  u.  H
) )  =  ( N `  H ) )
157, 14sseqtrd 3386 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  H )
)
16 mreexexlem2d.8 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  u.  H
)  e.  I )
1716adantr 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  e.  I
)
185, 3, 17mrissd 13863 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  X
)
1918unssbd 3527 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  X
)
203, 4, 19mrcssidd 13852 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( N `  H )
)
2115, 20unssd 3525 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  ( N `  H )
)
22 ssun2 3513 . . . . . . . . . 10  |-  H  C_  ( F  u.  H
)
2322a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( F  u.  H )
)
243, 4, 5, 21, 23, 17mrissmrcd 13867 . . . . . . . 8  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  =  H )
25 ssequn1 3519 . . . . . . . 8  |-  ( F 
C_  H  <->  ( F  u.  H )  =  H )
2624, 25sylibr 205 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  H
)
27 mreexexlem2d.5 . . . . . . . 8  |-  ( ph  ->  F  C_  ( X  \  H ) )
2827adantr 453 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( X  \  H ) )
2926, 28ssind 3567 . . . . . 6  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( H  i^i  ( X  \  H ) ) )
30 disjdif 3702 . . . . . 6  |-  ( H  i^i  ( X  \  H ) )  =  (/)
3129, 30syl6sseq 3396 . . . . 5  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  (/) )
32 ss0b 3659 . . . . 5  |-  ( F 
C_  (/)  <->  F  =  (/) )
3331, 32sylib 190 . . . 4  |-  ( (
ph  /\  G  =  (/) )  ->  F  =  (/) )
34 mreexexlem3d.9 . . . 4  |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )
351, 33, 34mpjaodan 763 . . 3  |-  ( ph  ->  F  =  (/) )
36 0elpw 4371 . . 3  |-  (/)  e.  ~P G
3735, 36syl6eqel 2526 . 2  |-  ( ph  ->  F  e.  ~P G
)
382elfvexd 5761 . . . 4  |-  ( ph  ->  X  e.  _V )
3927difss2d 3479 . . . 4  |-  ( ph  ->  F  C_  X )
4038, 39ssexd 4352 . . 3  |-  ( ph  ->  F  e.  _V )
41 enrefg 7141 . . 3  |-  ( F  e.  _V  ->  F  ~~  F )
4240, 41syl 16 . 2  |-  ( ph  ->  F  ~~  F )
43 breq2 4218 . . . 4  |-  ( i  =  F  ->  ( F  ~~  i  <->  F  ~~  F ) )
44 uneq1 3496 . . . . 5  |-  ( i  =  F  ->  (
i  u.  H )  =  ( F  u.  H ) )
4544eleq1d 2504 . . . 4  |-  ( i  =  F  ->  (
( i  u.  H
)  e.  I  <->  ( F  u.  H )  e.  I
) )
4643, 45anbi12d 693 . . 3  |-  ( i  =  F  ->  (
( F  ~~  i  /\  ( i  u.  H
)  e.  I )  <-> 
( F  ~~  F  /\  ( F  u.  H
)  e.  I ) ) )
4746rspcev 3054 . 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 1183 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 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {csn 3816   class class class wbr 4214   ` cfv 5456    ~~ cen 7108  Moorecmre 13809  mrClscmrc 13810  mrIndcmri 13811
This theorem is referenced by:  mreexexlem4d  13874  mreexexd  13875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-en 7112  df-mre 13813  df-mrc 13814  df-mri 13815
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