Theorem chle0 30961

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 30742	. 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		shle0 30960	. 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 2104   ⊆ wss 3949   S_ℋ csh 30446   C_ℋ cch 30447  0_ℋc0h 30453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-hilex 30517

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7416  df-sh 30725  df-ch 30739  df-ch0 30771

This theorem is referenced by:  chle0i  30970  chssoc  31014  hatomistici  31880  atcvat4i  31915