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Theorem ismred2 13820
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 2435 . . . 4  |-  (/)  =  (/)
3 rint0 4082 . . . 4  |-  ( (/)  =  (/)  ->  ( X  i^i  |^| (/) )  =  X )
42, 3ax-mp 8 . . 3  |-  ( X  i^i  |^| (/) )  =  X
5 0ss 3648 . . . 4  |-  (/)  C_  C
6 0ex 4331 . . . . 5  |-  (/)  e.  _V
7 sseq1 3361 . . . . . . 7  |-  ( s  =  (/)  ->  ( s 
C_  C  <->  (/)  C_  C
) )
87anbi2d 685 . . . . . 6  |-  ( s  =  (/)  ->  ( (
ph  /\  s  C_  C )  <->  ( ph  /\  (/)  C_  C ) ) )
9 inteq 4045 . . . . . . . 8  |-  ( s  =  (/)  ->  |^| s  =  |^| (/) )
109ineq2d 3534 . . . . . . 7  |-  ( s  =  (/)  ->  ( X  i^i  |^| s )  =  ( X  i^i  |^| (/) ) )
1110eleq1d 2501 . . . . . 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 2998 . . . 4  |-  ( (
ph  /\  (/)  C_  C
)  ->  ( X  i^i  |^| (/) )  e.  C
)
155, 14mpan2 653 . . 3  |-  ( ph  ->  ( X  i^i  |^| (/) )  e.  C )
164, 15syl5eqelr 2520 . 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 3350 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  C_  ~P X
)
20 simp3 959 . . . 4  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  -> 
s  =/=  (/) )
21 rintn0 4173 . . . 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 2510 . 2  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
251, 16, 24ismred 13819 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   |^|cint 4042   ` cfv 5446  Moorecmre 13799
This theorem is referenced by:  isacs1i  13874  mreacs  13875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-mre 13803
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