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Mirrors > Home > MPE Home > Th. List > 1stctop | Structured version Visualization version GIF version |
Description: A first-countable topology is a topology. (Contributed by Jeff Hankins, 22-Aug-2009.) |
Ref | Expression |
---|---|
1stctop | ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | is1stc 22343 | . 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧))))) |
3 | 2 | simplbi 501 | 1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 ∩ cin 3870 𝒫 cpw 4518 ∪ cuni 4824 class class class wbr 5058 ωcom 7649 ≼ cdom 8629 Topctop 21795 1stωc1stc 22339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-in 3878 df-ss 3888 df-pw 4520 df-uni 4825 df-1stc 22341 |
This theorem is referenced by: 1stcfb 22347 1stcrest 22355 1stcelcls 22363 lly1stc 22398 1stckgen 22456 tx1stc 22552 |
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