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Theorem ch0pss 22132
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 2602 . . 3  |-  ( 0H  =/=  A  <->  A  =/=  0H )
2 ch0le 22128 . . . 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 3244 . 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 1710    =/= wne 2521    C_ wss 3228    C. wpss 3229   CHcch 21617   0Hc0h 21623
This theorem is referenced by:  elat2  23028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-hilex 21687
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-xp 4774  df-cnv 4776  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fv 5342  df-ov 5945  df-sh 21894  df-ch 21909  df-ch0 21940
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