MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mreexexlem3d Structured version   Visualization version   Unicode version

Theorem mreexexlem3d 15630
Description: Base case of the induction in mreexexd 15632. (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 468 . . . 4  |-  ( (
ph  /\  F  =  (/) )  ->  F  =  (/) )
2 mreexexlem2d.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  (Moore `  X ) )
32adantr 472 . . . . . . . . 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 472 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  ( G  u.  H ) ) )
8 simpr 468 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  G  =  (/) )  ->  G  =  (/) )
98uneq1d 3578 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  (
(/)  u.  H )
)
10 uncom 3569 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  (
(/)  u.  H )
11 un0 3762 . . . . . . . . . . . . . 14  |-  ( H  u.  (/) )  =  H
1210, 11eqtr3i 2495 . . . . . . . . . . . . 13  |-  ( (/)  u.  H )  =  H
139, 12syl6eq 2521 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( G  u.  H )  =  H )
1413fveq2d 5883 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  ( N `  ( G  u.  H
) )  =  ( N `  H ) )
157, 14sseqtrd 3454 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( N `  H )
)
16 mreexexlem2d.8 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  u.  H
)  e.  I )
1716adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  e.  I
)
185, 3, 17mrissd 15620 . . . . . . . . . . . 12  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  X
)
1918unssbd 3603 . . . . . . . . . . 11  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  X
)
203, 4, 19mrcssidd 15609 . . . . . . . . . 10  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( N `  H )
)
2115, 20unssd 3601 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  C_  ( N `  H )
)
22 ssun2 3589 . . . . . . . . . 10  |-  H  C_  ( F  u.  H
)
2322a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  G  =  (/) )  ->  H  C_  ( F  u.  H )
)
243, 4, 5, 21, 23, 17mrissmrcd 15624 . . . . . . . 8  |-  ( (
ph  /\  G  =  (/) )  ->  ( F  u.  H )  =  H )
25 ssequn1 3595 . . . . . . . 8  |-  ( F 
C_  H  <->  ( F  u.  H )  =  H )
2624, 25sylibr 217 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  H
)
27 mreexexlem2d.5 . . . . . . . 8  |-  ( ph  ->  F  C_  ( X  \  H ) )
2827adantr 472 . . . . . . 7  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( X  \  H ) )
2926, 28ssind 3647 . . . . . 6  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  ( H  i^i  ( X  \  H ) ) )
30 disjdif 3830 . . . . . 6  |-  ( H  i^i  ( X  \  H ) )  =  (/)
3129, 30syl6sseq 3464 . . . . 5  |-  ( (
ph  /\  G  =  (/) )  ->  F  C_  (/) )
32 ss0b 3767 . . . . 5  |-  ( F 
C_  (/)  <->  F  =  (/) )
3331, 32sylib 201 . . . 4  |-  ( (
ph  /\  G  =  (/) )  ->  F  =  (/) )
34 mreexexlem3d.9 . . . 4  |-  ( ph  ->  ( F  =  (/)  \/  G  =  (/) ) )
351, 33, 34mpjaodan 803 . . 3  |-  ( ph  ->  F  =  (/) )
36 0elpw 4570 . . 3  |-  (/)  e.  ~P G
3735, 36syl6eqel 2557 . 2  |-  ( ph  ->  F  e.  ~P G
)
382elfvexd 5907 . . . 4  |-  ( ph  ->  X  e.  _V )
3927difss2d 3552 . . . 4  |-  ( ph  ->  F  C_  X )
4038, 39ssexd 4543 . . 3  |-  ( ph  ->  F  e.  _V )
41 enrefg 7619 . . 3  |-  ( F  e.  _V  ->  F  ~~  F )
4240, 41syl 17 . 2  |-  ( ph  ->  F  ~~  F )
43 breq2 4399 . . . 4  |-  ( i  =  F  ->  ( F  ~~  i  <->  F  ~~  F ) )
44 uneq1 3572 . . . . 5  |-  ( i  =  F  ->  (
i  u.  H )  =  ( F  u.  H ) )
4544eleq1d 2533 . . . 4  |-  ( i  =  F  ->  (
( i  u.  H
)  e.  I  <->  ( F  u.  H )  e.  I
) )
4643, 45anbi12d 725 . . 3  |-  ( i  =  F  ->  (
( F  ~~  i  /\  ( i  u.  H
)  e.  I )  <-> 
( F  ~~  F  /\  ( F  u.  H
)  e.  I ) ) )
4746rspcev 3136 . 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 1290 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 375    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   class class class wbr 4395   ` cfv 5589    ~~ cen 7584  Moorecmre 15566  mrClscmrc 15567  mrIndcmri 15568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-en 7588  df-mre 15570  df-mrc 15571  df-mri 15572
This theorem is referenced by:  mreexexlem4d  15631  mreexexd  15632
  Copyright terms: Public domain W3C validator