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Theorem ldlfcntref 30049
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 30048 . 2 Ldlf = CovHasRef{𝑥𝑥 ≼ ω}
3 vex 3234 . . . 4 𝑣 ∈ V
4 breq1 4688 . . . 4 (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
53, 4elab 3382 . . 3 (𝑣 ∈ {𝑥𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
65biimpi 206 . 2 (𝑣 ∈ {𝑥𝑥 ≼ ω} → 𝑣 ≼ ω)
71, 2, 6crefdf 30043 1 ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  wss 3607  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  ωcom 7107  cdom 7995  Refcref 21353  Ldlfcldlf 30047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-cref 30038  df-ldlf 30048
This theorem is referenced by: (None)
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