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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 30048 | . 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω} | |
3 | vex 3234 | . . . 4 ⊢ 𝑣 ∈ V | |
4 | breq1 4688 | . . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω)) | |
5 | 3, 4 | elab 3382 | . . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω) |
6 | 5 | biimpi 206 | . 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω) |
7 | 1, 2, 6 | crefdf 30043 | 1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 {cab 2637 ∃wrex 2942 ⊆ wss 3607 𝒫 cpw 4191 ∪ cuni 4468 class class class wbr 4685 ωcom 7107 ≼ cdom 7995 Refcref 21353 Ldlfcldlf 30047 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-cref 30038 df-ldlf 30048 |
This theorem is referenced by: (None) |
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