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Theorem ldlfcntref 32772
Description: Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.)
Hypothesis
Ref Expression
ldlfcntref.x 𝑋 = 𝐽
Assertion
Ref Expression
ldlfcntref ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
Distinct variable groups:   𝑣,𝐽   𝑣,𝑈
Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem ldlfcntref
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ldlfcntref.x . 2 𝑋 = 𝐽
2 df-ldlf 32771 . 2 Ldlf = CovHasRef{𝑥𝑥 ≼ ω}
3 vex 3479 . . . 4 𝑣 ∈ V
4 breq1 5150 . . . 4 (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
53, 4elab 3667 . . 3 (𝑣 ∈ {𝑥𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
65biimpi 215 . 2 (𝑣 ∈ {𝑥𝑥 ≼ ω} → 𝑣 ≼ ω)
71, 2, 6crefdf 32766 1 ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  wss 3947  𝒫 cpw 4601   cuni 4907   class class class wbr 5147  ωcom 7850  cdom 8933  Refcref 22988  Ldlfcldlf 32770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-cref 32761  df-ldlf 32771
This theorem is referenced by: (None)
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