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Theorem isclati 14497
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 519 . . 3  |-  ( s 
C_  B  ->  (
( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) )
54ax-gen 1552 . 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 14493 . 2  |-  ( K  e.  CLat  <->  ( K  e. 
Poset  /\  A. s ( s  C_  B  ->  ( ( U `  s
)  e.  B  /\  ( G `  s )  e.  B ) ) ) )
101, 5, 9mpbir2an 887 1  |-  K  e. 
CLat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721    C_ wss 3280   ` cfv 5413   Basecbs 13424   Posetcpo 14352   lubclub 14354   glbcglb 14355   CLatccla 14491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-clat 14492
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