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Theorem ch0pss 22904
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 2652 . . 3  |-  ( 0H  =/=  A  <->  A  =/=  0H )
2 ch0le 22900 . . . 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 3300 . 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 1721    =/= wne 2571    C_ wss 3284    C. wpss 3285   CHcch 22389   0Hc0h 22395
This theorem is referenced by:  elat2  23800
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-hilex 22459
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fv 5425  df-ov 6047  df-sh 22666  df-ch 22681  df-ch0 22712
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