Theorem ch0psst 9369

Description: The zero subspace is a proper subset of non-zero Hilbert lattice elements.

Assertion

Ref	Expression
ch0psst	|- (A e. CH -> (0H (. A <-> A =/= 0H))

Proof of Theorem ch0psst

Step	Hyp	Ref	Expression
1		ch0let 9365	. . . 4 |- (A e. CH -> 0H (_ A)
2	1	biantrurd 727	. . 3 |- (A e. CH -> (0H =/= A <-> (0H (_ A /\ 0H =/= A)))
3		necom 1636	. . 3 |- (0H =/= A <-> A =/= 0H)
4	2, 3	syl5bbr 534	. 2 |- (A e. CH -> (A =/= 0H <-> (0H (_ A /\ 0H =/= A)))
5		df-pss 2055	. 2 |- (0H (. A <-> (0H (_ A /\ 0H =/= A))
6	4, 5	syl6rbbr 539	1 |- (A e. CH -> (0H (. A <-> A =/= 0H))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958   =/= wne 1585   (_ wss 2047   (. wpss 2048  CHcch 8798  0Hc0h 8804

This theorem is referenced by:  elat2 10267

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-ne 1587  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-pss 2055  df-sn 2412  df-sh 9076  df-ch 9092  df-ch0 9125

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