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Theorem isatliN 30100
Description: Properties that determine an atomic lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
isatlati.1  |-  K  e. 
Lat
isatlati.b  |-  B  =  ( Base `  K
)
isatlati.l  |-  .<_  =  ( le `  K )
isatlati.z  |-  .0.  =  ( 0. `  K )
isatlati.a  |-  A  =  ( Atoms `  K )
isatlati.6  |-  .0.  e.  B
isatlati.7  |-  ( ( x  e.  B  /\  x  =/=  .0.  )  ->  E. y  e.  A  y  .<_  x )
Assertion
Ref Expression
isatliN  |-  K  e. 
AtLat
Distinct variable groups:    y, A    x, B    x, y, K
Allowed substitution hints:    A( x)    B( y)   
.<_ ( x, y)    .0. ( x, y)

Proof of Theorem isatliN
StepHypRef Expression
1 isatlati.1 . 2  |-  K  e. 
Lat
2 isatlati.6 . 2  |-  .0.  e.  B
3 isatlati.7 . . . 4  |-  ( ( x  e.  B  /\  x  =/=  .0.  )  ->  E. y  e.  A  y  .<_  x )
43ex 424 . . 3  |-  ( x  e.  B  ->  (
x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) )
54rgen 2771 . 2  |-  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x )
6 isatlati.b . . 3  |-  B  =  ( Base `  K
)
7 isatlati.l . . 3  |-  .<_  =  ( le `  K )
8 isatlati.z . . 3  |-  .0.  =  ( 0. `  K )
9 isatlati.a . . 3  |-  A  =  ( Atoms `  K )
106, 7, 8, 9isatl 30097 . 2  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) ) )
111, 2, 5, 10mpbir3an 1136 1  |-  K  e. 
AtLat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   0.cp0 14466   Latclat 14474   Atomscatm 30061   AtLatcal 30062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-atl 30096
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