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Mirrors > Home > MPE Home > Th. List > haustop | Structured version Visualization version GIF version |
Description: A Hausdorff space is a topology. (Contributed by NM, 5-Mar-2007.) |
Ref | Expression |
---|---|
haustop | ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ishaus 20936 | . 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))) |
3 | 2 | simplbi 475 | 1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ∅c0 3874 ∪ cuni 4372 Topctop 20517 Hauscha 20922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-uni 4373 df-haus 20929 |
This theorem is referenced by: haust1 20966 resthaus 20982 sshaus 20989 lmmo 20994 hauscmplem 21019 hauscmp 21020 hauslly 21105 hausllycmp 21107 kgenhaus 21157 pthaus 21251 txhaus 21260 xkohaus 21266 haushmph 21405 cmphaushmeo 21413 hausflim 21595 hauspwpwf1 21601 hauspwpwdom 21602 hausflf 21611 cnextfun 21678 cnextfvval 21679 cnextf 21680 cnextcn 21681 cnextfres1 21682 cnextfres 21683 qtophaus 29231 ismntop 29398 poimirlem30 32609 hausgraph 36809 |
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