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Theorem chle0 28642
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 28421 . 2 (𝐴C𝐴S )
2 shle0 28641 . 2 (𝐴S → (𝐴 ⊆ 0𝐴 = 0))
31, 2syl 17 1 (𝐴C → (𝐴 ⊆ 0𝐴 = 0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  wss 3723   S csh 28125   C cch 28126  0c0h 28132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-hilex 28196
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fv 6038  df-ov 6799  df-sh 28404  df-ch 28418  df-ch0 28450
This theorem is referenced by:  chle0i  28651  chssoc  28695  hatomistici  29561  atcvat4i  29596
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