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 31117 | . 2 ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω} | |
3 | vex 3497 | . . . 4 ⊢ 𝑣 ∈ V | |
4 | breq1 5069 | . . . 4 ⊢ (𝑥 = 𝑣 → (𝑥 ≼ ω ↔ 𝑣 ≼ ω)) | |
5 | 3, 4 | elab 3667 | . . 3 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} ↔ 𝑣 ≼ ω) |
6 | 5 | biimpi 218 | . 2 ⊢ (𝑣 ∈ {𝑥 ∣ 𝑥 ≼ ω} → 𝑣 ≼ ω) |
7 | 1, 2, 6 | crefdf 31112 | 1 ⊢ ((𝐽 ∈ Ldlf ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ≼ ω ∧ 𝑣Ref𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2799 ∃wrex 3139 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 class class class wbr 5066 ωcom 7580 ≼ cdom 8507 Refcref 22110 Ldlfcldlf 31116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-cref 31107 df-ldlf 31117 |
This theorem is referenced by: (None) |
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