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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 2823 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | is1stc 22051 | . 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 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 𝒫 cpw 4541 ∪ cuni 4840 class class class wbr 5068 ωcom 7582 ≼ cdom 8509 Topctop 21503 1stωc1stc 22047 |
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 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-in 3945 df-ss 3954 df-pw 4543 df-uni 4841 df-1stc 22049 |
This theorem is referenced by: 1stcfb 22055 1stcrest 22063 1stcelcls 22071 lly1stc 22106 1stckgen 22164 tx1stc 22260 |
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