Theorem chm1i 29239

Description: Meet with lattice one in C_ℋ. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ch0le.1	⊢ 𝐴 ∈ Cℋ

Assertion

Ref	Expression
chm1i	⊢ (𝐴 ∩ ℋ) = 𝐴

Proof of Theorem chm1i

Step	Hyp	Ref	Expression
1		ch0le.1	. . 3 ⊢ 𝐴 ∈ Cℋ
2	1	chssii 29014	. 2 ⊢ 𝐴 ⊆ ℋ
3		df-ss 3898	. 2 ⊢ (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴)
4	2, 3	mpbi 233	1 ⊢ (𝐴 ∩ ℋ) = 𝐴

Syntax hints:   = wceq 1538   ∈ wcel 2111   ∩ cin 3880   ⊆ wss 3881   ℋchba 28702   C_ℋ cch 28712

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-hilex 28782

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332  df-ov 7138  df-sh 28990  df-ch 29004

This theorem is referenced by:  stcltrlem1  30059