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Theorem chle0 29524
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 29305 . 2 (𝐴C𝐴S )
2 shle0 29523 . 2 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
31, 2syl 17 1 (𝐴C → (𝐴 ⊆ 0𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wss 3866   S csh 29009   C cch 29010  0c0h 29016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-hilex 29080
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fv 6388  df-ov 7216  df-sh 29288  df-ch 29302  df-ch0 29334
This theorem is referenced by:  chle0i  29533  chssoc  29577  hatomistici  30443  atcvat4i  30478
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