Theorem chle0 29805

Description: No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
chle0	⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))

Proof of Theorem chle0

Step	Hyp	Ref	Expression
1		chsh 29586	. 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		shle0 29804	. 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887   S_ℋ csh 29290   C_ℋ cch 29291  0_ℋc0h 29297

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-hilex 29361

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fv 6441  df-ov 7278  df-sh 29569  df-ch 29583  df-ch0 29615

This theorem is referenced by:  chle0i  29814  chssoc  29858  hatomistici  30724  atcvat4i  30759