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

Theorem isatliN 29785
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 2731 . 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 29782 . 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 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   0.cp0 14421   Latclat 14429   Atomscatm 29746   AtLatcal 29747
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-ne 2569  df-ral 2671  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-atl 29781
  Copyright terms: Public domain W3C validator