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Theorem chm1i 31259
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 31034 . 2 𝐴 ⊆ ℋ
3 df-ss 3961 . 2 (𝐴 ⊆ ℋ ↔ (𝐴 ∩ ℋ) = 𝐴)
42, 3mpbi 229 1 (𝐴 ∩ ℋ) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  cin 3943  wss 3944  chba 30722   C cch 30732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-hilex 30802
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fv 6550  df-ov 7417  df-sh 31010  df-ch 31024
This theorem is referenced by:  stcltrlem1  32079
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