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Theorem chle0 30388
Description: No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)
Assertion
Ref Expression
chle0 (𝐴C → (𝐴 ⊆ 0𝐴 = 0))

Proof of Theorem chle0
StepHypRef Expression
1 chsh 30169 . 2 (𝐴C𝐴S )
2 shle0 30387 . 2 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
31, 2syl 17 1 (𝐴C → (𝐴 ⊆ 0𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wss 3911   S csh 29873   C cch 29874  0c0h 29880
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 2708  ax-sep 5257  ax-hilex 29944
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 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fv 6505  df-ov 7361  df-sh 30152  df-ch 30166  df-ch0 30198
This theorem is referenced by:  chle0i  30397  chssoc  30441  hatomistici  31307  atcvat4i  31342
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