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Theorem ismred 13520
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 2804 . . . . 5  |-  s  e. 
_V
43elpw 3644 . . . 4  |-  ( s  e.  ~P C  <->  s  C_  C )
5 ismred.in . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
653expia 1153 . . . 4  |-  ( (
ph  /\  s  C_  C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
74, 6sylan2b 461 . . 3  |-  ( (
ph  /\  s  e.  ~P C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
87ralrimiva 2639 . 2  |-  ( ph  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )
9 ismre 13508 . 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 1136 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   |^|cint 3878   ` cfv 5271  Moorecmre 13500
This theorem is referenced by:  ismred2  13521  mremre  13522  submre  13523  subrgmre  15585  lssmre  15739  cssmre  16609  cldmre  16831  toponmre  16846  ismrcd1  26876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-mre 13504
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