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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldlfcntref | Structured version Visualization version GIF version |
Description: Every open cover of a Lindelöf space has a countable refinement. (Contributed by Thierry Arnoux, 1-Feb-2020.) |
Ref | Expression |
---|---|
ldlfcntref.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ldlfcntref | ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldlfcntref.x | . 2 ⊢ 𝑋 = ∪ 𝐽 | |
2 | df-ldlf 32771 | . 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω} | |
3 | vex 3479 | . . . 4 ⊢ 𝑣 ∈ V | |
4 | breq1 5150 | . . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω)) | |
5 | 3, 4 | elab 3667 | . . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω) |
6 | 5 | biimpi 215 | . 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω) |
7 | 1, 2, 6 | crefdf 32766 | 1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {cab 2710 ∃wrex 3071 ⊆ wss 3947 𝒫 cpw 4601 ∪ cuni 4907 class class class wbr 5147 ωcom 7850 ≼ cdom 8933 Refcref 22988 Ldlfcldlf 32770 |
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 5298 |
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 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-cref 32761 df-ldlf 32771 |
This theorem is referenced by: (None) |
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