Theorem chle0 28995

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 28774	. 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		shle0 28994	. 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1507   ∈ wcel 2050   ⊆ wss 3823   S_ℋ csh 28478   C_ℋ cch 28479  0_ℋc0h 28485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744  ax-sep 5054  ax-hilex 28549

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-xp 5407  df-cnv 5409  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fv 6190  df-ov 6973  df-sh 28757  df-ch 28771  df-ch0 28803

This theorem is referenced by:  chle0i  29004  chssoc  29048  hatomistici  29914  atcvat4i  29949