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Theorem ismred 15019
Description: Properties that determine a Moore collection. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Hypotheses
Ref Expression
ismred.ss  |-  ( ph  ->  C  C_  ~P X
)
ismred.ba  |-  ( ph  ->  X  e.  C )
ismred.in  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
Assertion
Ref Expression
ismred  |-  ( ph  ->  C  e.  (Moore `  X ) )
Distinct variable groups:    ph, s    C, s    X, s

Proof of Theorem ismred
StepHypRef Expression
1 ismred.ss . 2  |-  ( ph  ->  C  C_  ~P X
)
2 ismred.ba . 2  |-  ( ph  ->  X  e.  C )
3 selpw 4022 . . . 4  |-  ( s  e.  ~P C  <->  s  C_  C )
4 ismred.in . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
543expia 1198 . . . 4  |-  ( (
ph  /\  s  C_  C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
63, 5sylan2b 475 . . 3  |-  ( (
ph  /\  s  e.  ~P C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
76ralrimiva 2871 . 2  |-  ( ph  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )
8 ismre 15007 . 2  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
91, 2, 7, 8syl3anbrc 1180 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    e. wcel 1819    =/= wne 2652   A.wral 2807    C_ wss 3471   (/)c0 3793   ~Pcpw 4015   |^|cint 4288   ` cfv 5594  Moorecmre 14999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-mre 15003
This theorem is referenced by:  ismred2  15020  mremre  15021  submre  15022  subrgmre  17580  lssmre  17739  cssmre  18851  cldmre  19706  toponmre  19721  ismrcd1  30835
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