Theorem chnlen0 5369

Description: A Hilbert lattice element that is not a subset of another is nonzero.

Assertion

Ref	Expression
chnlen0	|- (B e. CH -> (-. A (_ B -> -. A = 0H))

Proof of Theorem chnlen0

Step	Hyp	Ref	Expression
1		sseq1 1521	. . . 4 |- (A = 0H -> (A (_ B <-> 0H (_ B))
2		ch0let 5366	. . . 4 |- (B e. CH -> 0H (_ B)
3	1, 2	syl5bir 184	. . 3 |- (A = 0H -> (B e. CH -> A (_ B))
4	3	com12 13	. 2 |- (B e. CH -> (A = 0H -> A (_ B))
5	4	con3d 87	1 |- (B e. CH -> (-. A (_ B -> -. A = 0H))

Syntax hints:  -. wn 1   -> wi 2   = wceq 1091   e. wcel 1092   (_ wss 1487  CHcch 4968  0Hc0h 4974

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

metamath.org