Theorem chm0i 31577

Description: Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ch0le.1	⊢ 𝐴 ∈ Cℋ

Assertion

Ref	Expression
chm0i	⊢ (𝐴 ∩ 0ℋ) = 0ℋ

Proof of Theorem chm0i

Step	Hyp	Ref	Expression
1		inss2 4192	. 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ
2		ch0le.1	. . . 4 ⊢ 𝐴 ∈ Cℋ
3	2	ch0lei 31538	. . 3 ⊢ 0ℋ ⊆ 𝐴
4		ssid 3958	. . 3 ⊢ 0ℋ ⊆ 0ℋ
5	3, 4	ssini 4194	. 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ)
6	1, 5	eqssi 3952	1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ

Syntax hints:   = wceq 1542   ∈ wcel 2114   ∩ cin 3902   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:  chm0  31578