Theorem ldlfcntref 32834

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 32833	. 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
3		vex 3479	. . . 4 ⊢ 𝑣 ∈ V
4		breq1 5152	. . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
5	3, 4	elab 3669	. . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
6	5	biimpi 215	. 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω)
7	1, 2, 6	crefdf 32828	1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909   class class class wbr 5149  ωcom 7855   ≼ cdom 8937  Refcref 23006  Ldlfcldlf 32832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-cref 32823  df-ldlf 32833

This theorem is referenced by: (None)