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

Theorem chm1i 28299
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 28072 . 2 𝐴 ⊆ ℋ
3 df-ss 3586 . 2 (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴)
42, 3mpbi 220 1 (𝐴 ∩ ℋ) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  wcel 1989  cin 3571  wss 3572  chil 27760   C cch 27770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-hilex 27840
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fv 5894  df-ov 6650  df-sh 28048  df-ch 28062
This theorem is referenced by:  stcltrlem1  29119
  Copyright terms: Public domain W3C validator