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Theorem ldlfcntref 29055
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 29054 . 2 Ldlf = CovHasRef{𝑥𝑥 ≼ ω}
3 vex 3175 . . . 4 𝑣 ∈ V
4 breq1 4580 . . . 4 (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
53, 4elab 3318 . . 3 (𝑣 ∈ {𝑥𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
65biimpi 204 . 2 (𝑣 ∈ {𝑥𝑥 ≼ ω} → 𝑣 ≼ ω)
71, 2, 6crefdf 29049 1 ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2595  wrex 2896  wss 3539  𝒫 cpw 4107   cuni 4366   class class class wbr 4577  ωcom 6934  cdom 7816  Refcref 21057  Ldlfcldlf 29053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-cref 29044  df-ldlf 29054
This theorem is referenced by: (None)
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