Theorem chnlen0 9363

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 2085	. . 3 |- (A = 0H -> (A (_ B <-> 0H (_ B))
2		ch0let 9360	. . 3 |- (B e. CH -> 0H (_ B)
3	1, 2	syl5cbir 211	. 2 |- (B e. CH -> (A = 0H -> A (_ B))
4	3	con3d 95	1 |- (B e. CH -> (-. A (_ B -> -. A = 0H))

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   e. wcel 960   (_ wss 2050  CHcch 8793  0Hc0h 8799

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2054  df-ss 2056  df-sn 2416  df-sh 9071  df-ch 9087  df-ch0 9120

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