Theorem chm1i 31786

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 31561	. 2 ⊢ 𝐴 ⊆ ℋ
3		dfss2 3924	. 2 ⊢ (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴)
4	2, 3	mpbi 233	1 ⊢ (𝐴 ∩ ℋ) = 𝐴

Syntax hints:   = wceq 1570   ∈ wcel 2143   ∩ cin 3905   ⊆ wss 3906   ℋchba 31249   C_ℋ cch 31259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-hilex 31329

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fv 6546  df-ov 7415  df-sh 31537  df-ch 31551

This theorem is referenced by:  stcltrlem1  32606