Theorem chnlen0 31425

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 31422	. . 3 ⊢ (𝐵 ∈ Cℋ → 0ℋ ⊆ 𝐵)
2		sseq1 3984	. . 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 1540   ∈ wcel 2108   ⊆ wss 3926   C_ℋ cch 30910  0_ℋc0h 30916

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-hilex 30980

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fv 6539  df-ov 7408  df-sh 31188  df-ch 31202  df-ch0 31234

This theorem is referenced by: (None)