Theorem chm0i 29880

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 4166	. 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ
2		ch0le.1	. . . 4 ⊢ 𝐴 ∈ Cℋ
3	2	ch0lei 29841	. . 3 ⊢ 0ℋ ⊆ 𝐴
4		ssid 3945	. . 3 ⊢ 0ℋ ⊆ 0ℋ
5	3, 4	ssini 4168	. 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ)
6	1, 5	eqssi 3939	1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ

Syntax hints:   = wceq 1537   ∈ wcel 2101   ∩ cin 3888   C_ℋ cch 29319  0_ℋc0h 29325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-sep 5226  ax-hilex 29389

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-xp 5597  df-cnv 5599  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fv 6455  df-ov 7298  df-sh 29597  df-ch 29611  df-ch0 29643

This theorem is referenced by:  chm0  29881