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Theorem atlex 29799
 Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 23816 analog.) (Contributed by NM, 21-Oct-2011.)
Hypotheses
Ref Expression
atlex.b
atlex.l
atlex.z
atlex.a
Assertion
Ref Expression
atlex
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem atlex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 atlex.b . . . . 5
2 atlex.l . . . . 5
3 atlex.z . . . . 5
4 atlex.a . . . . 5
51, 2, 3, 4isatl 29782 . . . 4
65simp3bi 974 . . 3
7 neeq1 2575 . . . . 5
8 breq2 4176 . . . . . 6
98rexbidv 2687 . . . . 5
107, 9imbi12d 312 . . . 4
1110rspccv 3009 . . 3
126, 11syl 16 . 2
13123imp 1147 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 936   wceq 1649   wcel 1721   wne 2567  wral 2666  wrex 2667   class class class wbr 4172  cfv 5413  cbs 13424  cple 13491  cp0 14421  clat 14429  catm 29746  cal 29747 This theorem is referenced by:  atnle  29800  atlatmstc  29802  cvratlem  29903  cvrat4  29925  2llnmat  30006  2lnat  30266 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
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