Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hlhgt4 Structured version   Unicode version

Theorem hlhgt4 30247
 Description: A Hilbert lattice has a height of at least 4. (Contributed by NM, 4-Dec-2011.)
Hypotheses
Ref Expression
hlhgt4.b
hlhgt4.s
hlhgt4.z
hlhgt4.u
Assertion
Ref Expression
hlhgt4
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)

Proof of Theorem hlhgt4
StepHypRef Expression
1 hlhgt4.b . . 3
2 eqid 2438 . . 3
3 hlhgt4.s . . 3
4 eqid 2438 . . 3
5 hlhgt4.z . . 3
6 hlhgt4.u . . 3
7 eqid 2438 . . 3
81, 2, 3, 4, 5, 6, 7ishlat2 30213 . 2
9 simprr 735 . 2
108, 9sylbi 189 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726   wne 2601  wral 2707  wrex 2708   class class class wbr 4214  cfv 5456  (class class class)co 6083  cbs 13471  cple 13538  cplt 14400  cjn 14403  cp0 14468  cp1 14469  ccla 14538  coml 30035  catm 30123  cal 30124  chlt 30210 This theorem is referenced by:  hlhgt2  30248  athgt  30315 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-cvlat 30182  df-hlat 30211
 Copyright terms: Public domain W3C validator