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Theorem ch0pss 29149
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 necom 3066 . . 3 (0𝐴𝐴 ≠ 0)
2 ch0le 29145 . . . 4 (𝐴C → 0𝐴)
32biantrurd 533 . . 3 (𝐴C → (0𝐴 ↔ (0𝐴 ∧ 0𝐴)))
41, 3syl5bbr 286 . 2 (𝐴C → (𝐴 ≠ 0 ↔ (0𝐴 ∧ 0𝐴)))
5 df-pss 3951 . 2 (0𝐴 ↔ (0𝐴 ∧ 0𝐴))
64, 5syl6rbbr 291 1 (𝐴C → (0𝐴𝐴 ≠ 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wne 3013  wss 3933  wpss 3934   C cch 28633  0c0h 28639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7148  df-sh 28911  df-ch 28925  df-ch0 28957
This theorem is referenced by:  elat2  30044
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