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

Theorem isatliN 29492
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 2608 . 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 29489 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   Latclat 14151   Atomscatm 29453   AtLatcal 29454
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-ne 2448  df-ral 2548  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-atl 29488
  Copyright terms: Public domain W3C validator