Theorem chle0 11000

Description: No Hilbert lattice element is smaller than zero.

Assertion

Ref	Expression
chle0	|- (A e. CH -> (A C_ 0H <-> A = 0H))

Proof of Theorem chle0

Step	Hyp	Ref	Expression
1		chsh 10729	. 2 |- (A e. CH -> A e. SH)
2		shle0 10999	. 2 |- (A e. SH -> (A C_ 0H <-> A = 0H))
3	1, 2	syl 12	1 |- (A e. CH -> (A C_ 0H <-> A = 0H))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300   C_ wss 2593  SHcsh 10429  CHcch 10430  0Hc0h 10436

This theorem is referenced by:  chle0i 11008  chssoc 11052  hatomistici 11934  atcvat4i 11969

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-v 2294  df-in 2603  df-ss 2605  df-sn 3049  df-sh 10709  df-ch 10725  df-ch0 10758

Copyright terms: Public domain