Theorem chm0 5411

Description: Meet with Hilbert lattice zero.

Hypothesis

Ref	Expression
ch0le.1	⊢ A ∈ Cℋ

Assertion

Ref	Expression
chm0	⊢ (A ∩ 0ℋ) = 0ℋ

 

Proof of Theorem chm0

Step	Hyp	Ref	Expression
1		inss2 1658	. 2 ⊢ (A ∩ 0ℋ) ⊆ 0ℋ
2	ch0le.1	. . . 4 ⊢ A ∈ Cℋ
3	2	ch0le  5373	. . 3 ⊢ 0ℋ ⊆ A
4		ssid 1519	. . 3 ⊢ 0ℋ ⊆ 0ℋ
5	3, 4	ssini 1660	. 2 ⊢ 0ℋ ⊆ (A ∩ 0ℋ)
6	1, 5	eqssi 1517	1 ⊢ (A ∩ 0ℋ) = 0ℋ

Syntax hints:   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   C_ℋ cch 4968  0_ℋc0h 4974

This theorem is referenced by:  fh1 5518  fh2 5519

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sn 1811  df-sh 5114  df-ch 5127  df-ch0 5157

metamath.org