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Mirrors > Home > MPE Home > Th. List > xrhaus | Structured version Visualization version GIF version |
Description: The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
Ref | Expression |
---|---|
xrhaus | ⊢ (ordTop‘ ≤ ) ∈ Haus |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | letsr 17955 | . 2 ⊢ ≤ ∈ TosetRel | |
2 | ordthaus 22137 | . 2 ⊢ ( ≤ ∈ TosetRel → (ordTop‘ ≤ ) ∈ Haus) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (ordTop‘ ≤ ) ∈ Haus |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ‘cfv 6339 ≤ cle 10756 ordTopcordt 16877 TosetRel ctsr 17927 Hauscha 22061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-pre-lttri 10691 ax-pre-lttrn 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-om 7602 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-fi 8950 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-topgen 16822 df-ordt 16879 df-ps 17928 df-tsr 17929 df-top 21647 df-topon 21664 df-bases 21699 df-haus 22068 |
This theorem is referenced by: xrge0haus 31468 esumpfinval 31615 esumpfinvalf 31616 xlimuni 42958 xlimfun 42960 |
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