Theorem ldlfcntref 34210

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 34209	. 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω}
3		vex 3457	. . . 4 ⊢ 𝑣 ∈ V
4		breq1 5111	. . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω))
5	3, 4	elab 3637	. . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω)
6	5	biimpi 219	. 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω)
7	1, 2, 6	crefdf 34204	1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  {cab 2739  ∃wrex 3087   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871   class class class wbr 5108  ωcom 7861   ≼ cdom 8940  Refcref 23638  Ldlfcldlf 34208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-cref 34199  df-ldlf 34209

This theorem is referenced by: (None)