Theorem chm0i 30743

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 4230	. 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ
2		ch0le.1	. . . 4 ⊢ 𝐴 ∈ Cℋ
3	2	ch0lei 30704	. . 3 ⊢ 0ℋ ⊆ 𝐴
4		ssid 4005	. . 3 ⊢ 0ℋ ⊆ 0ℋ
5	3, 4	ssini 4232	. 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ)
6	1, 5	eqssi 3999	1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ

Syntax hints:   = wceq 1542   ∈ wcel 2107   ∩ cin 3948   C_ℋ cch 30182  0_ℋc0h 30188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-hilex 30252

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-sh 30460  df-ch 30474  df-ch0 30506

This theorem is referenced by:  chm0  30744