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Theorem mreexexlem3d 15262
Description: Base case of the induction in mreexexd 15264. (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 461 . . . 4  |-  ( (
ph  /\  F  =  (/) )  ->  F  =  (/) )
2 mreexexlem2d.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  (Moore `  X ) )
32adantr 465 . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  ( G  u.  H ) ) )
8 simpr 461 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  G  =  (/) )  ->  G  =  (/) )
98uneq1d 3598 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  (
(/)  u.  H )
)
10 uncom 3589 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  (
(/)  u.  H )
11 un0 3766 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  H
1210, 11eqtr3i 2435 . . . . . . . . . . . . 13  |-  ( (/)  u.  H )  =  H
139, 12syl6eq 2461 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  H )
1413fveq2d 5855 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  ( N `  ( G  u.  H
) )  =  ( N `  H ) )
157, 14sseqtrd 3480 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  H )
)
16 mreexexlem2d.8 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  u.  H
)  e.  I )
1716adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  e.  I
)
185, 3, 17mrissd 15252 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  X
)
1918unssbd 3623 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  X
)
203, 4, 19mrcssidd 15241 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( N `  H )
)
2115, 20unssd 3621 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  ( N `  H )
)
22 ssun2 3609 . . . . . . . . . 10  |-  H  C_  ( F  u.  H
)
2322a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( F  u.  H )
)
243, 4, 5, 21, 23, 17mrissmrcd 15256 . . . . . . . 8  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  =  H )
25 ssequn1 3615 . . . . . . . 8  |-  ( F 
C_  H  <->  ( F  u.  H )  =  H )
2624, 25sylibr 214 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  H
)
27 mreexexlem2d.5 . . . . . . . 8  |-  ( ph  ->  F  C_  ( X  \  H ) )
2827adantr 465 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( X  \  H ) )
2926, 28ssind 3665 . . . . . 6  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( H  i^i  ( X  \  H ) ) )
30 disjdif 3846 . . . . . 6  |-  ( H  i^i  ( X  \  H ) )  =  (/)
3129, 30syl6sseq 3490 . . . . 5  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  (/) )
32 ss0b 3771 . . . . 5  |-  ( F 
C_  (/)  <->  F  =  (/) )
3331, 32sylib 198 . . . 4  |-  ( (
ph  /\  G  =  (/) )  ->  F  =  (/) )
34 mreexexlem3d.9 . . . 4  |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )
351, 33, 34mpjaodan 789 . . 3  |-  ( ph  ->  F  =  (/) )
36 0elpw 4565 . . 3  |-  (/)  e.  ~P G
3735, 36syl6eqel 2500 . 2  |-  ( ph  ->  F  e.  ~P G
)
382elfvexd 5879 . . . 4  |-  ( ph  ->  X  e.  _V )
3927difss2d 3575 . . . 4  |-  ( ph  ->  F  C_  X )
4038, 39ssexd 4543 . . 3  |-  ( ph  ->  F  e.  _V )
41 enrefg 7587 . . 3  |-  ( F  e.  _V  ->  F  ~~  F )
4240, 41syl 17 . 2  |-  ( ph  ->  F  ~~  F )
43 breq2 4401 . . . 4  |-  ( i  =  F  ->  ( F  ~~  i  <->  F  ~~  F ) )
44 uneq1 3592 . . . . 5  |-  ( i  =  F  ->  (
i  u.  H )  =  ( F  u.  H ) )
4544eleq1d 2473 . . . 4  |-  ( i  =  F  ->  (
( i  u.  H
)  e.  I  <->  ( F  u.  H )  e.  I
) )
4643, 45anbi12d 711 . . 3  |-  ( i  =  F  ->  (
( F  ~~  i  /\  ( i  u.  H
)  e.  I )  <-> 
( F  ~~  F  /\  ( F  u.  H
)  e.  I ) ) )
4746rspcev 3162 . 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 1230 1  |-  ( ph  ->  E. i  e.  ~P  G ( F  ~~  i  /\  ( i  u.  H )  e.  I
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756   E.wrex 2757   _Vcvv 3061    \ cdif 3413    u. cun 3414    i^i cin 3415    C_ wss 3416   (/)c0 3740   ~Pcpw 3957   {csn 3974   class class class wbr 4397   ` cfv 5571    ~~ cen 7553  Moorecmre 15198  mrClscmrc 15199  mrIndcmri 15200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-int 4230  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-en 7557  df-mre 15202  df-mrc 15203  df-mri 15204
This theorem is referenced by:  mreexexlem4d  15263  mreexexd  15264
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