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Theorem ismred2 13756
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 2388 . . . 4  |-  (/)  =  (/)
3 rint0 4033 . . . 4  |-  ( (/)  =  (/)  ->  ( X  i^i  |^| (/) )  =  X )
42, 3ax-mp 8 . . 3  |-  ( X  i^i  |^| (/) )  =  X
5 0ss 3600 . . . 4  |-  (/)  C_  C
6 0ex 4281 . . . . 5  |-  (/)  e.  _V
7 sseq1 3313 . . . . . . 7  |-  ( s  =  (/)  ->  ( s 
C_  C  <->  (/)  C_  C
) )
87anbi2d 685 . . . . . 6  |-  ( s  =  (/)  ->  ( (
ph  /\  s  C_  C )  <->  ( ph  /\  (/)  C_  C ) ) )
9 inteq 3996 . . . . . . . 8  |-  ( s  =  (/)  ->  |^| s  =  |^| (/) )
109ineq2d 3486 . . . . . . 7  |-  ( s  =  (/)  ->  ( X  i^i  |^| s )  =  ( X  i^i  |^| (/) ) )
1110eleq1d 2454 . . . . . 6  |-  ( s  =  (/)  ->  ( ( X  i^i  |^| s
)  e.  C  <->  ( X  i^i  |^| (/) )  e.  C
) )
128, 11imbi12d 312 . . . . 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 2950 . . . 4  |-  ( (
ph  /\  (/)  C_  C
)  ->  ( X  i^i  |^| (/) )  e.  C
)
155, 14mpan2 653 . . 3  |-  ( ph  ->  ( X  i^i  |^| (/) )  e.  C )
164, 15syl5eqelr 2473 . 2  |-  ( ph  ->  X  e.  C )
17 simp2 958 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  C )
1813ad2ant1 978 . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  C  C_  ~P X )
1917, 18sstrd 3302 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  ~P X
)
20 simp3 959 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  =/=  (/) )
21 rintn0 4123 . . . 4  |-  ( ( s  C_  ~P X  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  =  |^| s )
2219, 20, 21syl2anc 643 . . 3  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  =  |^| s )
23133adant3 977 . . 3  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
( X  i^i  |^| s )  e.  C
)
2422, 23eqeltrrd 2463 . 2  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
251, 16, 24ismred 13755 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551    i^i cin 3263    C_ wss 3264   (/)c0 3572   ~Pcpw 3743   |^|cint 3993   ` cfv 5395  Moorecmre 13735
This theorem is referenced by:  isacs1i  13810  mreacs  13811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-mre 13739
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