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Theorem atlex 29506
Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 22940 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atlex.b  |-  B  =  ( Base `  K
)
atlex.l  |-  .<_  =  ( le `  K )
atlex.z  |-  .0.  =  ( 0. `  K )
atlex.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atlex  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  E. y  e.  A  y  .<_  X )
Distinct variable groups:    y, A    y, K    y, X
Allowed substitution hints:    B( y)    .<_ ( y)    .0. ( y)

Proof of Theorem atlex
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 atlex.b . . . . 5  |-  B  =  ( Base `  K
)
2 atlex.l . . . . 5  |-  .<_  =  ( le `  K )
3 atlex.z . . . . 5  |-  .0.  =  ( 0. `  K )
4 atlex.a . . . . 5  |-  A  =  ( Atoms `  K )
51, 2, 3, 4isatl 29489 . . . 4  |-  ( K  e.  AtLat 
<->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x ) ) )
65simp3bi 972 . . 3  |-  ( K  e.  AtLat  ->  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) )
7 neeq1 2454 . . . . 5  |-  ( x  =  X  ->  (
x  =/=  .0.  <->  X  =/=  .0.  ) )
8 breq2 4027 . . . . . 6  |-  ( x  =  X  ->  (
y  .<_  x  <->  y  .<_  X ) )
98rexbidv 2564 . . . . 5  |-  ( x  =  X  ->  ( E. y  e.  A  y  .<_  x  <->  E. y  e.  A  y  .<_  X ) )
107, 9imbi12d 311 . . . 4  |-  ( x  =  X  ->  (
( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x )  <->  ( X  =/=  .0.  ->  E. y  e.  A  y  .<_  X ) ) )
1110rspccv 2881 . . 3  |-  ( A. x  e.  B  (
x  =/=  .0.  ->  E. y  e.  A  y 
.<_  x )  ->  ( X  e.  B  ->  ( X  =/=  .0.  ->  E. y  e.  A  y 
.<_  X ) ) )
126, 11syl 15 . 2  |-  ( K  e.  AtLat  ->  ( X  e.  B  ->  ( X  =/=  .0.  ->  E. y  e.  A  y  .<_  X ) ) )
13123imp 1145 1  |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  X  =/= 
.0.  )  ->  E. y  e.  A  y  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = 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 is referenced by:  atnle  29507  atlatmstc  29509  cvratlem  29610  cvrat4  29632  2llnmat  29713  2lnat  29973
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
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