Theorem chle0 31252

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 31033	. 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		shle0 31251	. 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 1534   ∈ wcel 2099   ⊆ wss 3947   S_ℋ csh 30737   C_ℋ cch 30738  0_ℋc0h 30744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-hilex 30808

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fv 6556  df-ov 7423  df-sh 31016  df-ch 31030  df-ch0 31062

This theorem is referenced by:  chle0i  31261  chssoc  31305  hatomistici  32171  atcvat4i  32206