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Theorem ldlfcntref 33512
Description: Every open cover of a LindelΓΆf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
Hypothesis
Ref Expression
ldlfcntref.x 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ldlfcntref ((𝐽 ∈ Ldlf ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 β‰Ό Ο‰ ∧ 𝑣Refπ‘ˆ))
Distinct variable groups:   𝑣,𝐽   𝑣,π‘ˆ
Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem ldlfcntref
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ldlfcntref.x . 2 𝑋 = βˆͺ 𝐽
2 df-ldlf 33511 . 2 Ldlf = CovHasRef{π‘₯ ∣ π‘₯ β‰Ό Ο‰}
3 vex 3467 . . . 4 𝑣 ∈ V
4 breq1 5146 . . . 4 (π‘₯ = 𝑣 β†’ (π‘₯ β‰Ό Ο‰ ↔ 𝑣 β‰Ό Ο‰))
53, 4elab 3659 . . 3 (𝑣 ∈ {π‘₯ ∣ π‘₯ β‰Ό Ο‰} ↔ 𝑣 β‰Ό Ο‰)
65biimpi 215 . 2 (𝑣 ∈ {π‘₯ ∣ π‘₯ β‰Ό Ο‰} β†’ 𝑣 β‰Ό Ο‰)
71, 2, 6crefdf 33506 1 ((𝐽 ∈ Ldlf ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 β‰Ό Ο‰ ∧ 𝑣Refπ‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆƒwrex 3060   βŠ† wss 3939  π’« cpw 4598  βˆͺ cuni 4903   class class class wbr 5143  Ο‰com 7868   β‰Ό cdom 8960  Refcref 23424  Ldlfcldlf 33510
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 2696  ax-sep 5294
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-cref 33501  df-ldlf 33511
This theorem is referenced by: (None)
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