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Theorem ldlfcntref 31207
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 31206 . 2 Ldlf = CovHasRef{𝑥𝑥 ≼ ω}
3 vex 3447 . . . 4 𝑣 ∈ V
4 breq1 5036 . . . 4 (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
53, 4elab 3618 . . 3 (𝑣 ∈ {𝑥𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
65biimpi 219 . 2 (𝑣 ∈ {𝑥𝑥 ≼ ω} → 𝑣 ≼ ω)
71, 2, 6crefdf 31201 1 ((𝐽 ∈ Ldlf ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  wss 3884  𝒫 cpw 4500   cuni 4803   class class class wbr 5033  ωcom 7564  cdom 8494  Refcref 22110  Ldlfcldlf 31205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-cref 31196  df-ldlf 31206
This theorem is referenced by: (None)
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