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Theorem ch0pss 22947
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 2685 . . 3  |-  ( 0H  =/=  A  <->  A  =/=  0H )
2 ch0le 22943 . . . 4  |-  ( A  e.  CH  ->  0H  C_  A )
32biantrurd 495 . . 3  |-  ( A  e.  CH  ->  ( 0H  =/=  A  <->  ( 0H  C_  A  /\  0H  =/=  A ) ) )
41, 3syl5bbr 251 . 2  |-  ( A  e.  CH  ->  ( A  =/=  0H  <->  ( 0H  C_  A  /\  0H  =/=  A ) ) )
5 df-pss 3336 . 2  |-  ( 0H 
C.  A  <->  ( 0H  C_  A  /\  0H  =/=  A ) )
64, 5syl6rbbr 256 1  |-  ( A  e.  CH  ->  ( 0H  C.  A  <->  A  =/=  0H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725    =/= wne 2599    C_ wss 3320    C. wpss 3321   CHcch 22432   0Hc0h 22438
This theorem is referenced by:  elat2  23843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-ov 6084  df-sh 22709  df-ch 22724  df-ch0 22755
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