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Theorem chle0 29793
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 29574 . 2 (𝐴C𝐴S )
2 shle0 29792 . 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 2110  wss 3892   S csh 29278   C cch 29279  0c0h 29285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-hilex 29349
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fv 6439  df-ov 7272  df-sh 29557  df-ch 29571  df-ch0 29603
This theorem is referenced by:  chle0i  29802  chssoc  29846  hatomistici  30712  atcvat4i  30747
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