Theorem ch0psst 9284

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 9280	. . . 4 |- (A e. CH -> 0H (_ A)
2	1	biantrurd 725	. . 3 |- (A e. CH -> (0H =/= A <-> (0H (_ A /\ 0H =/= A)))
3		necom 1628	. . 3 |- (0H =/= A <-> A =/= 0H)
4	2, 3	syl5bbr 532	. 2 |- (A e. CH -> (A =/= 0H <-> (0H (_ A /\ 0H =/= A)))
5		df-pss 2045	. 2 |- (0H (. A <-> (0H (_ A /\ 0H =/= A))
6	4, 5	syl6rbbr 537	1 |- (A e. CH -> (0H (. A <-> A =/= 0H))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 955   =/= wne 1577   (_ wss 2037   (. wpss 2038  CHcch 8737  0Hc0h 8743

This theorem is referenced by:  elat2 10175

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-in 2041  df-ss 2043  df-pss 2045  df-sn 2402  df-sh 8997  df-ch 9013  df-ch0 9046

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