Theorem chnlen0 9368

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 2082	. . 3 |- (A = 0H -> (A (_ B <-> 0H (_ B))
2		ch0let 9365	. . 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 956   e. wcel 958   (_ wss 2047  CHcch 8798  0Hc0h 8804

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-sn 2412  df-sh 9076  df-ch 9092  df-ch0 9125

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