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Theorem chnlen0 22787
Description: A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
chnlen0  |-  ( B  e.  CH  ->  ( -.  A  C_  B  ->  -.  A  =  0H ) )

Proof of Theorem chnlen0
StepHypRef Expression
1 ch0le 22784 . . 3  |-  ( B  e.  CH  ->  0H  C_  B )
2 sseq1 3305 . . 3  |-  ( A  =  0H  ->  ( A  C_  B  <->  0H  C_  B
) )
31, 2syl5ibrcom 214 . 2  |-  ( B  e.  CH  ->  ( A  =  0H  ->  A 
C_  B ) )
43con3d 127 1  |-  ( B  e.  CH  ->  ( -.  A  C_  B  ->  -.  A  =  0H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3256   CHcch 22273   0Hc0h 22279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-hilex 22343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fv 5395  df-ov 6016  df-sh 22550  df-ch 22565  df-ch0 22596
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