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Theorem mreexexlem3d 13564
Description: Base case of the induction in mreexexd 13566. (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 3341 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  (
(/)  u.  H )
)
10 uncom 3332 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  (
(/)  u.  H )
11 un0 3492 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  H
1210, 11eqtr3i 2318 . . . . . . . . . . . . 13  |-  ( (/)  u.  H )  =  H
139, 12syl6eq 2344 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  H )
1413fveq2d 5545 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  ( N `  ( G  u.  H
) )  =  ( N `  H ) )
157, 14sseqtrd 3227 . . . . . . . . . 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 13554 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  X
)
1918unssbd 3366 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  X
)
203, 4, 19mrcssidd 13543 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( N `  H )
)
2115, 20unssd 3364 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  ( N `  H )
)
22 ssun2 3352 . . . . . . . . . 10  |-  H  C_  ( F  u.  H
)
2322a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( F  u.  H )
)
243, 4, 5, 21, 23, 17mrissmrcd 13558 . . . . . . . 8  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  =  H )
25 ssequn1 3358 . . . . . . . 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 3406 . . . . . 6  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( H  i^i  ( X  \  H ) ) )
30 disjdif 3539 . . . . . 6  |-  ( H  i^i  ( X  \  H ) )  =  (/)
3129, 30syl6sseq 3237 . . . . 5  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  (/) )
32 ss0b 3497 . . . . 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 4196 . . 3  |-  (/)  e.  ~P G
3735, 36syl6eqel 2384 . 2  |-  ( ph  ->  F  e.  ~P G
)
382elfvexd 5572 . . . 4  |-  ( ph  ->  X  e.  _V )
3927difss2d 3319 . . . 4  |-  ( ph  ->  F  C_  X )
4038, 39ssexd 4177 . . 3  |-  ( ph  ->  F  e.  _V )
41 enrefg 6909 . . 3  |-  ( F  e.  _V  ->  F  ~~  F )
4240, 41syl 15 . 2  |-  ( ph  ->  F  ~~  F )
43 breq2 4043 . . . 4  |-  ( i  =  F  ->  ( F  ~~  i  <->  F  ~~  F ) )
44 uneq1 3335 . . . . 5  |-  ( i  =  F  ->  (
i  u.  H )  =  ( F  u.  H ) )
4544eleq1d 2362 . . . 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 2897 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   {csn 3653   class class class wbr 4039   ` cfv 5271    ~~ cen 6876  Moorecmre 13500  mrClscmrc 13501  mrIndcmri 13502
This theorem is referenced by:  mreexexlem4d  13565  mreexexd  13566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-en 6880  df-mre 13504  df-mrc 13505  df-mri 13506
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