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Theorem ch0pss 31474
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 3983 . 2 (0𝐴 ↔ (0𝐴 ∧ 0𝐴))
2 necom 2992 . . 3 (0𝐴𝐴 ≠ 0)
3 ch0le 31470 . . . 4 (𝐴C → 0𝐴)
43biantrurd 532 . . 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 206  wa 395  wcel 2106  wne 2938  wss 3963  wpss 3964   C cch 30958  0c0h 30964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-sh 31236  df-ch 31250  df-ch0 31282
This theorem is referenced by:  elat2  32369
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