MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismred Structured version   Unicode version

Theorem ismred 13828
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 vex 2960 . . . . 5  |-  s  e. 
_V
43elpw 3806 . . . 4  |-  ( s  e.  ~P C  <->  s  C_  C )
5 ismred.in . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
653expia 1156 . . . 4  |-  ( (
ph  /\  s  C_  C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
74, 6sylan2b 463 . . 3  |-  ( (
ph  /\  s  e.  ~P C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
87ralrimiva 2790 . 2  |-  ( ph  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )
9 ismre 13816 . 2  |-  ( C  e.  (Moore `  X
)  <->  ( C  C_  ~P X  /\  X  e.  C  /\  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C ) ) )
101, 2, 8, 9syl3anbrc 1139 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    e. wcel 1726    =/= wne 2600   A.wral 2706    C_ wss 3321   (/)c0 3629   ~Pcpw 3800   |^|cint 4051   ` cfv 5455  Moorecmre 13808
This theorem is referenced by:  ismred2  13829  mremre  13830  submre  13831  subrgmre  15893  lssmre  16043  cssmre  16921  cldmre  17143  toponmre  17158  ismrcd1  26753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-mre 13812
  Copyright terms: Public domain W3C validator