Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atl0cl Structured version   Unicode version

Theorem atl0cl 30002
 Description: An atomic lattice has a zero element. We can use this in place of op0cl 29883 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
atl0cl.b
atl0cl.z
Assertion
Ref Expression
atl0cl

Proof of Theorem atl0cl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 atl0cl.b . . 3
2 eqid 2435 . . 3
3 atl0cl.z . . 3
4 eqid 2435 . . 3
51, 2, 3, 4isatl 29998 . 2
65simp2bi 973 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698   class class class wbr 4204  cfv 5446  cbs 13459  cple 13526  cp0 14456  clat 14464  catm 29962  cal 29963 This theorem is referenced by:  atl0le  30003  atlle0  30004  atlltn0  30005  isat3  30006  atnle0  30008  atlen0  30009  atcmp  30010  atcvreq0  30013  pmap0  30463  dia0  31751  dih0cnv  31982 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 29997
 Copyright terms: Public domain W3C validator