Theorem ch0psst 5370

Description: The zero subspace is a proper subset of non-zero Hilbert lattice elements.

Assertion

Ref	Expression
ch0psst	⊢ (A ∈ Cℋ → (0ℋ ⊂ A ↔ ¬ A = 0ℋ))

Proof of Theorem ch0psst

Step	Hyp	Ref	Expression
1		chle0t 5368	. . . 4 ⊢ (A ∈ Cℋ → (A ⊆ 0ℋ ↔ A = 0ℋ))
2	1	negbid 463	. . 3 ⊢ (A ∈ Cℋ → (¬ A ⊆ 0ℋ ↔ ¬ A = 0ℋ))
3		ch0let 5366	. . . 4 ⊢ (A ∈ Cℋ → 0ℋ ⊆ A)
4	3	biantrurd 546	. . 3 ⊢ (A ∈ Cℋ → (¬ A ⊆ 0ℋ ↔ (0ℋ ⊆ A ∧ ¬ A ⊆ 0ℋ)))
5	2, 4	bitr3d 408	. 2 ⊢ (A ∈ Cℋ → (¬ A = 0ℋ ↔ (0ℋ ⊆ A ∧ ¬ A ⊆ 0ℋ)))
6		dfpss3 1558	. 2 ⊢ (0ℋ ⊂ A ↔ (0ℋ ⊆ A ∧ ¬ A ⊆ 0ℋ))
7	5, 6	syl6rbbr 417	1 ⊢ (A ∈ Cℋ → (0ℋ ⊂ A ↔ ¬ A = 0ℋ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487   ⊂ wpss 1488   C_ℋ cch 4968  0_ℋc0h 4974

This theorem is referenced by:  elat2 5739

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-pss 1494  df-sn 1811  df-sh 5114  df-ch 5127  df-ch0 5157

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