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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 2824 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | is1stc 22052 | . 2 ⊢ (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∃𝑦 ∈ 𝒫 𝐽(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧))))) |
3 | 2 | simplbi 500 | 1 ⊢ (𝐽 ∈ 1stω → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2113 ∀wral 3141 ∃wrex 3142 ∩ cin 3938 𝒫 cpw 4542 ∪ cuni 4841 class class class wbr 5069 ωcom 7583 ≼ cdom 8510 Topctop 21504 1stωc1stc 22048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-in 3946 df-ss 3955 df-pw 4544 df-uni 4842 df-1stc 22050 |
This theorem is referenced by: 1stcfb 22056 1stcrest 22064 1stcelcls 22072 lly1stc 22107 1stckgen 22165 tx1stc 22261 |
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