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Theorem isclati 14219
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 1533 . 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 14215 . 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 1527    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255   Basecbs 13148   Posetcpo 14074   lubclub 14076   glbcglb 14077   CLatccla 14213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-clat 14214
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