Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isatliN Unicode version

Theorem isatliN 30114
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 423 . . 3  |-  ( x  e.  B  ->  (
x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) )
54rgen 2621 . 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 30111 . 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 1134 1  |-  K  e. 
AtLat
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   Latclat 14167   Atomscatm 30075   AtLatcal 30076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-atl 30110
  Copyright terms: Public domain W3C validator