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Theorem ismred2 14875
Description: Properties that determine a Moore collection, using restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Hypotheses
Ref Expression
ismred2.ss  |-  ( ph  ->  C  C_  ~P X
)
ismred2.in  |-  ( (
ph  /\  s  C_  C )  ->  ( X  i^i  |^| s )  e.  C )
Assertion
Ref Expression
ismred2  |-  ( ph  ->  C  e.  (Moore `  X ) )
Distinct variable groups:    ph, s    C, s    X, s

Proof of Theorem ismred2
StepHypRef Expression
1 ismred2.ss . 2  |-  ( ph  ->  C  C_  ~P X
)
2 eqid 2467 . . . 4  |-  (/)  =  (/)
3 rint0 4328 . . . 4  |-  ( (/)  =  (/)  ->  ( X  i^i  |^| (/) )  =  X )
42, 3ax-mp 5 . . 3  |-  ( X  i^i  |^| (/) )  =  X
5 0ss 3819 . . . 4  |-  (/)  C_  C
6 0ex 4583 . . . . 5  |-  (/)  e.  _V
7 sseq1 3530 . . . . . . 7  |-  ( s  =  (/)  ->  ( s 
C_  C  <->  (/)  C_  C
) )
87anbi2d 703 . . . . . 6  |-  ( s  =  (/)  ->  ( (
ph  /\  s  C_  C )  <->  ( ph  /\  (/)  C_  C ) ) )
9 inteq 4291 . . . . . . . 8  |-  ( s  =  (/)  ->  |^| s  =  |^| (/) )
109ineq2d 3705 . . . . . . 7  |-  ( s  =  (/)  ->  ( X  i^i  |^| s )  =  ( X  i^i  |^| (/) ) )
1110eleq1d 2536 . . . . . 6  |-  ( s  =  (/)  ->  ( ( X  i^i  |^| s
)  e.  C  <->  ( X  i^i  |^| (/) )  e.  C
) )
128, 11imbi12d 320 . . . . 5  |-  ( s  =  (/)  ->  ( ( ( ph  /\  s  C_  C )  ->  ( X  i^i  |^| s )  e.  C )  <->  ( ( ph  /\  (/)  C_  C )  ->  ( X  i^i  |^| (/) )  e.  C ) ) )
13 ismred2.in . . . . 5  |-  ( (
ph  /\  s  C_  C )  ->  ( X  i^i  |^| s )  e.  C )
146, 12, 13vtocl 3170 . . . 4  |-  ( (
ph  /\  (/)  C_  C
)  ->  ( X  i^i  |^| (/) )  e.  C
)
155, 14mpan2 671 . . 3  |-  ( ph  ->  ( X  i^i  |^| (/) )  e.  C )
164, 15syl5eqelr 2560 . 2  |-  ( ph  ->  X  e.  C )
17 simp2 997 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  C )
1813ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  C  C_  ~P X )
1917, 18sstrd 3519 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  ~P X
)
20 simp3 998 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  =/=  (/) )
21 rintn0 4422 . . . 4  |-  ( ( s  C_  ~P X  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  =  |^| s )
2219, 20, 21syl2anc 661 . . 3  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  =  |^| s )
23133adant3 1016 . . 3  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  e.  C
)
2422, 23eqeltrrd 2556 . 2  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
251, 16, 24ismred 14874 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   |^|cint 4288   ` cfv 5594  Moorecmre 14854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-mre 14858
This theorem is referenced by:  isacs1i  14929  mreacs  14930
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