Theorem chle0t 5368

Description: No Hilbert lattice element is smaller than zero.

Assertion

Ref	Expression
chle0t	|- (A e. CH -> (A (_ 0H <-> A = 0H))

Proof of Theorem chle0t

Step	Hyp	Ref	Expression
1		chsh 5131	. 2 |- (A e. CH -> A e. SH)
2		shle0t 5367	. 2 |- (A e. SH -> (A (_ 0H <-> A = 0H))
3	1, 2	syl 12	1 |- (A e. CH -> (A (_ 0H <-> A = 0H))

Syntax hints:   -> wi 2   <-> wb 127   = wceq 1091   e. wcel 1092   (_ wss 1487  SHcsh 4967  CHcch 4968  0Hc0h 4974

This theorem is referenced by:  ch0psst 5370  chle0 5374  hatomistic 5755  atcvat4 5775

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sn 1811  df-sh 5114  df-ch 5127  df-ch0 5157

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