Theorem chnlen0 31531

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 31528	. . 3 ⊢ (𝐵 ∈ Cℋ → 0ℋ ⊆ 𝐵)
2		sseq1 3961	. . 3 ⊢ (𝐴 = 0ℋ → (𝐴 ⊆ 𝐵 ↔ 0ℋ ⊆ 𝐵))
3	1, 2	syl5ibrcom 247	. 2 ⊢ (𝐵 ∈ Cℋ → (𝐴 = 0ℋ → 𝐴 ⊆ 𝐵))
4	3	con3d 152	1 ⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903   C_ℋ cch 31016  0_ℋc0h 31022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-hilex 31086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fv 6508  df-ov 7371  df-sh 31294  df-ch 31308  df-ch0 31340

This theorem is referenced by: (None)