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Theorem ch0pss 22024
Description: The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
ch0pss  |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H ) )

Proof of Theorem ch0pss
StepHypRef Expression
1 necom 2527 . . 3  |-  ( 0H  =/=  A  <->  A  =/=  0H )
2 ch0le 22020 . . . 4  |-  ( A  e.  CH  ->  0H  C_  A )
32biantrurd 494 . . 3  |-  ( A  e.  CH  ->  ( 0H  =/=  A  <->  ( 0H  C_  A  /\  0H  =/=  A ) ) )
41, 3syl5bbr 250 . 2  |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  ( 0H  C_  A  /\  0H  =/=  A ) ) )
5 df-pss 3168 . 2  |-  ( 0H 
C.  A  <->  ( 0H  C_  A  /\  0H  =/=  A ) )
64, 5syl6rbbr 255 1  |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684    =/= wne 2446    C_ wss 3152    C. wpss 3153   CHcch 21509   0Hc0h 21515
This theorem is referenced by:  elat2  22920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-sh 21786  df-ch 21801  df-ch0 21832
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