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Theorem ch0pss 29214
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 3067 . . 3 (0𝐴𝐴 ≠ 0)
2 ch0le 29210 . . . 4 (𝐴C → 0𝐴)
32biantrurd 535 . . 3 (𝐴C → (0𝐴 ↔ (0𝐴 ∧ 0𝐴)))
41, 3syl5bbr 287 . 2 (𝐴C → (𝐴 ≠ 0 ↔ (0𝐴 ∧ 0𝐴)))
5 df-pss 3952 . 2 (0𝐴 ↔ (0𝐴 ∧ 0𝐴))
64, 5syl6rbbr 292 1 (𝐴C → (0𝐴𝐴 ≠ 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2107  wne 3014  wss 3934  wpss 3935   C cch 28698  0c0h 28704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-hilex 28768
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 7151  df-sh 28976  df-ch 28990  df-ch0 29022
This theorem is referenced by:  elat2  30109
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