Theorem chm0i 31372

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 4228	. 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ
2		ch0le.1	. . . 4 ⊢ 𝐴 ∈ Cℋ
3	2	ch0lei 31333	. . 3 ⊢ 0ℋ ⊆ 𝐴
4		ssid 3999	. . 3 ⊢ 0ℋ ⊆ 0ℋ
5	3, 4	ssini 4230	. 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ)
6	1, 5	eqssi 3993	1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ

Syntax hints:   = wceq 1533   ∈ wcel 2098   ∩ cin 3943   C_ℋ cch 30811  0_ℋc0h 30817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-hilex 30881

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fv 6557  df-ov 7422  df-sh 31089  df-ch 31103  df-ch0 31135

This theorem is referenced by:  chm0  31373