Theorem chm0i 31576

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 4179	. 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ
2		ch0le.1	. . . 4 ⊢ 𝐴 ∈ Cℋ
3	2	ch0lei 31537	. . 3 ⊢ 0ℋ ⊆ 𝐴
4		ssid 3945	. . 3 ⊢ 0ℋ ⊆ 0ℋ
5	3, 4	ssini 4181	. 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ)
6	1, 5	eqssi 3939	1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ

Syntax hints:   = wceq 1542   ∈ wcel 2114   ∩ cin 3889   C_ℋ cch 31015  0_ℋc0h 31021

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 5231  ax-hilex 31085

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fv 6500  df-ov 7363  df-sh 31293  df-ch 31307  df-ch0 31339

This theorem is referenced by:  chm0  31577