Theorem ldlfcntref 31113

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 31112	. 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
3		vex 3497	. . . 4 ⊢ 𝑣 ∈ V
4		breq1 5061	. . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
5	3, 4	elab 3666	. . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
6	5	biimpi 218	. 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω)
7	1, 2, 6	crefdf 31107	1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799  ∃wrex 3139   ⊆ wss 3935  𝒫 cpw 4538  ∪ cuni 4831   class class class wbr 5058  ωcom 7574   ≼ cdom 8501  Refcref 22104  Ldlfcldlf 31111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-cref 31102  df-ldlf 31112

This theorem is referenced by: (None)