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 29920 | . 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω} | |
3 | vex 3203 | . . . 4 ⊢ 𝑣 ∈ V | |
4 | breq1 4656 | . . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω)) | |
5 | 3, 4 | elab 3350 | . . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω) |
6 | 5 | biimpi 206 | . 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω) |
7 | 1, 2, 6 | crefdf 29915 | 1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 class class class wbr 4653 ωcom 7065 ≼ cdom 7953 Refcref 21305 Ldlfcldlf 29919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-cref 29910 df-ldlf 29920 |
This theorem is referenced by: (None) |
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