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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 2818 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ishaus 21858 | . 2 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))) |
3 | 2 | simplbi 498 | 1 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ∅c0 4288 ∪ cuni 4830 Topctop 21429 Hauscha 21844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-uni 4831 df-haus 21851 |
This theorem is referenced by: haust1 21888 resthaus 21904 sshaus 21911 lmmo 21916 hauscmplem 21942 hauscmp 21943 hauslly 22028 hausllycmp 22030 kgenhaus 22080 pthaus 22174 txhaus 22183 xkohaus 22189 haushmph 22328 cmphaushmeo 22336 hausflim 22517 hauspwpwf1 22523 hauspwpwdom 22524 hausflf 22533 cnextfun 22600 cnextfvval 22601 cnextf 22602 cnextcn 22603 cnextfres1 22604 cnextfres 22605 qtophaus 30999 ismntop 31166 poimirlem30 34803 hausgraph 39690 |
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