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Theorem atlex 30041
 Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 23855 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 30024 . . . 4
65simp3bi 974 . . 3
7 neeq1 2606 . . . . 5
8 breq2 4208 . . . . . 6
98rexbidv 2718 . . . . 5
107, 9imbi12d 312 . . . 4
1110rspccv 3041 . . 3
126, 11syl 16 . 2
13123imp 1147 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 936   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698   class class class wbr 4204  cfv 5446  cbs 13461  cple 13528  cp0 14458  clat 14466  catm 29988  cal 29989 This theorem is referenced by:  atnle  30042  atlatmstc  30044  cvratlem  30145  cvrat4  30167  2llnmat  30248  2lnat  30508 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-atl 30023
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