Theorem chnlen0 31796

Description: A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
chnlen0	⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ))

Proof of Theorem chnlen0

Step	Hyp	Ref	Expression
1		ch0le 31793	. . 3 ⊢ (𝐵 ∈ Cℋ → 0ℋ ⊆ 𝐵)
2		sseq1 3962	. . 3 ⊢ (𝐴 = 0ℋ → (𝐴 ⊆ 𝐵 ↔ 0ℋ ⊆ 𝐵))
3	1, 2	syl5ibrcom 250	. 2 ⊢ (𝐵 ∈ Cℋ → (𝐴 = 0ℋ → 𝐴 ⊆ 𝐵))
4	3	con3d 153	1 ⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905   C_ℋ cch 31281  0_ℋc0h 31287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-hilex 31351

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fv 6544  df-ov 7413  df-sh 31559  df-ch 31573  df-ch0 31605

This theorem is referenced by: (None)