Theorem chm1i 29719

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 29494	. 2 ⊢ 𝐴 ⊆ ℋ
3		df-ss 3900	. 2 ⊢ (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴)
4	2, 3	mpbi 229	1 ⊢ (𝐴 ∩ ℋ) = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883   ℋchba 29182   C_ℋ cch 29192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fv 6426  df-ov 7258  df-sh 29470  df-ch 29484

This theorem is referenced by:  stcltrlem1  30539