Theorem ldlfcntref 29249

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 ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 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.

Step	Hyp	Ref	Expression
1		ldlfcntref.x	. 2 ⊢ 𝑋 = ∪ 𝐽
2		df-ldlf 29248	. 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
3		vex 3176	. . . 4 ⊢ 𝑣 ∈ V
4		breq1 4586	. . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
5	3, 4	elab 3319	. . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
6	5	biimpi 205	. 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω)
7	1, 2, 6	crefdf 29243	1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ωcom 6957   ≼ cdom 7839  Refcref 21115  Ldlfcldlf 29247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-cref 29238  df-ldlf 29248

This theorem is referenced by: (None)