HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chm1i Structured version   Visualization version   GIF version

Theorem chm1i 31204
Description: Meet with lattice one in C. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
ch0le.1 𝐴C
Assertion
Ref Expression
chm1i (𝐴 ∩ ℋ) = 𝐴

Proof of Theorem chm1i
StepHypRef Expression
1 ch0le.1 . . 3 𝐴C
21chssii 30979 . 2 𝐴 ⊆ ℋ
3 df-ss 3958 . 2 (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴)
42, 3mpbi 229 1 (𝐴 ∩ ℋ) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  cin 3940  wss 3941  chba 30667   C cch 30677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-hilex 30747
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fv 6542  df-ov 7405  df-sh 30955  df-ch 30969
This theorem is referenced by:  stcltrlem1  32024
  Copyright terms: Public domain W3C validator