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Theorem ismred 14639
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 3962 . . . 4  |-  ( s  e.  ~P C  <->  s  C_  C )
4 ismred.in . . . . 5  |-  ( (
ph  /\  s  C_  C  /\  s  =/=  (/) )  ->  |^| s  e.  C
)
543expia 1190 . . . 4  |-  ( (
ph  /\  s  C_  C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
63, 5sylan2b 475 . . 3  |-  ( (
ph  /\  s  e.  ~P C )  ->  (
s  =/=  (/)  ->  |^| s  e.  C ) )
76ralrimiva 2820 . 2  |-  ( ph  ->  A. s  e.  ~P  C ( s  =/=  (/)  ->  |^| s  e.  C
) )
8 ismre 14627 . 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 1172 1  |-  ( ph  ->  C  e.  (Moore `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    e. wcel 1758    =/= wne 2642   A.wral 2793    C_ wss 3423   (/)c0 3732   ~Pcpw 3955   |^|cint 4223   ` cfv 5513  Moorecmre 14619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-mre 14623
This theorem is referenced by:  ismred2  14640  mremre  14641  submre  14642  subrgmre  16992  lssmre  17150  cssmre  18224  cldmre  18795  toponmre  18810  ismrcd1  29169
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