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Theorem ismred2 14546
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 2443 . . . 4  |-  (/)  =  (/)
3 rint0 4173 . . . 4  |-  ( (/)  =  (/)  ->  ( X  i^i  |^| (/) )  =  X )
42, 3ax-mp 5 . . 3  |-  ( X  i^i  |^| (/) )  =  X
5 0ss 3671 . . . 4  |-  (/)  C_  C
6 0ex 4427 . . . . 5  |-  (/)  e.  _V
7 sseq1 3382 . . . . . . 7  |-  ( s  =  (/)  ->  ( s 
C_  C  <->  (/)  C_  C
) )
87anbi2d 703 . . . . . 6  |-  ( s  =  (/)  ->  ( (
ph  /\  s  C_  C )  <->  ( ph  /\  (/)  C_  C ) ) )
9 inteq 4136 . . . . . . . 8  |-  ( s  =  (/)  ->  |^| s  =  |^| (/) )
109ineq2d 3557 . . . . . . 7  |-  ( s  =  (/)  ->  ( X  i^i  |^| s )  =  ( X  i^i  |^| (/) ) )
1110eleq1d 2509 . . . . . 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 3029 . . . 4  |-  ( (
ph  /\  (/)  C_  C
)  ->  ( X  i^i  |^| (/) )  e.  C
)
155, 14mpan2 671 . . 3  |-  ( ph  ->  ( X  i^i  |^| (/) )  e.  C )
164, 15syl5eqelr 2528 . 2  |-  ( ph  ->  X  e.  C )
17 simp2 989 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  C )
1813ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  C  C_  ~P X )
1917, 18sstrd 3371 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  ~P X
)
20 simp3 990 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  =/=  (/) )
21 rintn0 4266 . . . 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 1008 . . 3  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  e.  C
)
2422, 23eqeltrrd 2518 . 2  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
251, 16, 24ismred 14545 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611    i^i cin 3332    C_ wss 3333   (/)c0 3642   ~Pcpw 3865   |^|cint 4133   ` cfv 5423  Moorecmre 14525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-mre 14529
This theorem is referenced by:  isacs1i  14600  mreacs  14601
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