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Theorem ch0pss 30698
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 (𝐴C → (0𝐴𝐴 ≠ 0))

Proof of Theorem ch0pss
StepHypRef Expression
1 df-pss 3968 . 2 (0𝐴 ↔ (0𝐴 ∧ 0𝐴))
2 necom 2995 . . 3 (0𝐴𝐴 ≠ 0)
3 ch0le 30694 . . . 4 (𝐴C → 0𝐴)
43biantrurd 534 . . 3 (𝐴C → (0𝐴 ↔ (0𝐴 ∧ 0𝐴)))
52, 4bitr3id 285 . 2 (𝐴C → (𝐴 ≠ 0 ↔ (0𝐴 ∧ 0𝐴)))
61, 5bitr4id 290 1 (𝐴C → (0𝐴𝐴 ≠ 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wne 2941  wss 3949  wpss 3950   C cch 30182  0c0h 30188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-hilex 30252
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-sh 30460  df-ch 30474  df-ch0 30506
This theorem is referenced by:  elat2  31593
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