Theorem ldlfcntref 33998

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 33997	. 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
3		vex 3433	. . . 4 ⊢ 𝑣 ∈ V
4		breq1 5088	. . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
5	3, 4	elab 3622	. . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
6	5	biimpi 216	. 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω)
7	1, 2, 6	crefdf 33992	1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714  ∃wrex 3061   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850   class class class wbr 5085  ωcom 7817   ≼ cdom 8891  Refcref 23467  Ldlfcldlf 33996

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-cref 33987  df-ldlf 33997

This theorem is referenced by: (None)