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Theorem ismred2 15557
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 2461 . . . 4  |-  (/)  =  (/)
3 rint0 4288 . . . 4  |-  ( (/)  =  (/)  ->  ( X  i^i  |^| (/) )  =  X )
42, 3ax-mp 5 . . 3  |-  ( X  i^i  |^| (/) )  =  X
5 0ss 3774 . . . 4  |-  (/)  C_  C
6 0ex 4548 . . . . 5  |-  (/)  e.  _V
7 sseq1 3464 . . . . . . 7  |-  ( s  =  (/)  ->  ( s 
C_  C  <->  (/)  C_  C
) )
87anbi2d 715 . . . . . 6  |-  ( s  =  (/)  ->  ( (
ph  /\  s  C_  C )  <->  ( ph  /\  (/)  C_  C ) ) )
9 inteq 4250 . . . . . . . 8  |-  ( s  =  (/)  ->  |^| s  =  |^| (/) )
109ineq2d 3645 . . . . . . 7  |-  ( s  =  (/)  ->  ( X  i^i  |^| s )  =  ( X  i^i  |^| (/) ) )
1110eleq1d 2523 . . . . . 6  |-  ( s  =  (/)  ->  ( ( X  i^i  |^| s
)  e.  C  <->  ( X  i^i  |^| (/) )  e.  C
) )
128, 11imbi12d 326 . . . . 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 3111 . . . 4  |-  ( (
ph  /\  (/)  C_  C
)  ->  ( X  i^i  |^| (/) )  e.  C
)
155, 14mpan2 682 . . 3  |-  ( ph  ->  ( X  i^i  |^| (/) )  e.  C )
164, 15syl5eqelr 2544 . 2  |-  ( ph  ->  X  e.  C )
17 simp2 1015 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  C )
1813ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  C  C_  ~P X )
1917, 18sstrd 3453 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  ~P X
)
20 simp3 1016 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  =/=  (/) )
21 rintn0 4385 . . . 4  |-  ( ( s  C_  ~P X  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  =  |^| s )
2219, 20, 21syl2anc 671 . . 3  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  =  |^| s )
23133adant3 1034 . . 3  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  e.  C
)
2422, 23eqeltrrd 2540 . 2  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
251, 16, 24ismred 15556 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632    i^i cin 3414    C_ wss 3415   (/)c0 3742   ~Pcpw 3962   |^|cint 4247   ` cfv 5600  Moorecmre 15536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-int 4248  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5564  df-fun 5602  df-fv 5608  df-mre 15540
This theorem is referenced by:  isacs1i  15611  mreacs  15612
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