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Theorem isclati 14318
Description: Properties that determine a complete lattice. (Contributed by NM, 12-Sep-2011.)
Hypotheses
Ref Expression
isclati.1  |-  K  e. 
Poset
isclati.b  |-  B  =  ( Base `  K
)
isclati.u  |-  U  =  ( lub `  K
)
isclati.g  |-  G  =  ( glb `  K
)
isclati.5  |-  ( s 
C_  B  ->  ( U `  s )  e.  B )
isclati.6  |-  ( s 
C_  B  ->  ( G `  s )  e.  B )
Assertion
Ref Expression
isclati  |-  K  e. 
CLat
Distinct variable group:    K, s
Allowed substitution hints:    B( s)    U( s)    G( s)

Proof of Theorem isclati
StepHypRef Expression
1 isclati.1 . 2  |-  K  e. 
Poset
2 isclati.5 . . . 4  |-  ( s 
C_  B  ->  ( U `  s )  e.  B )
3 isclati.6 . . . 4  |-  ( s 
C_  B  ->  ( G `  s )  e.  B )
42, 3jca 518 . . 3  |-  ( s 
C_  B  ->  (
( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) )
54ax-gen 1546 . 2  |-  A. s
( s  C_  B  ->  ( ( U `  s )  e.  B  /\  ( G `  s
)  e.  B ) )
6 isclati.b . . 3  |-  B  =  ( Base `  K
)
7 isclati.u . . 3  |-  U  =  ( lub `  K
)
8 isclati.g . . 3  |-  G  =  ( glb `  K
)
96, 7, 8isclat 14314 . 2  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) ) ) )
101, 5, 9mpbir2an 886 1  |-  K  e. 
CLat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1540    = wceq 1642    e. wcel 1710    C_ wss 3228   ` cfv 5337   Basecbs 13245   Posetcpo 14173   lubclub 14175   glbcglb 14176   CLatccla 14312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-clat 14313
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