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Mirrors > Home > MPE Home > Th. List > haust1 | Structured version Visualization version Unicode version |
Description: A Hausdorff space is a T1 space. (Contributed by FL, 11-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
haust1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . . . 9 | |
2 | 1 | hausnei 21132 | . . . . . . . 8 |
3 | simprr1 1109 | . . . . . . . . . . 11 | |
4 | noel 3919 | . . . . . . . . . . . . 13 | |
5 | simprr3 1111 | . . . . . . . . . . . . . 14 | |
6 | 5 | eleq2d 2687 | . . . . . . . . . . . . 13 |
7 | 4, 6 | mtbiri 317 | . . . . . . . . . . . 12 |
8 | simprr2 1110 | . . . . . . . . . . . . 13 | |
9 | elin 3796 | . . . . . . . . . . . . . 14 | |
10 | 9 | simplbi2com 657 | . . . . . . . . . . . . 13 |
11 | 8, 10 | syl 17 | . . . . . . . . . . . 12 |
12 | 7, 11 | mtod 189 | . . . . . . . . . . 11 |
13 | 3, 12 | jca 554 | . . . . . . . . . 10 |
14 | 13 | rexlimdvaa 3032 | . . . . . . . . 9 |
15 | 14 | reximdva 3017 | . . . . . . . 8 |
16 | 2, 15 | mpd 15 | . . . . . . 7 |
17 | rexanali 2998 | . . . . . . 7 | |
18 | 16, 17 | sylib 208 | . . . . . 6 |
19 | 18 | 3exp2 1285 | . . . . 5 |
20 | 19 | imp32 449 | . . . 4 |
21 | 20 | necon4ad 2813 | . . 3 |
22 | 21 | ralrimivva 2971 | . 2 |
23 | haustop 21135 | . . . 4 | |
24 | 1 | toptopon 20722 | . . . 4 TopOn |
25 | 23, 24 | sylib 208 | . . 3 TopOn |
26 | ist1-2 21151 | . . 3 TopOn | |
27 | 25, 26 | syl 17 | . 2 |
28 | 22, 27 | mpbird 247 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 wral 2912 wrex 2913 cin 3573 c0 3915 cuni 4436 cfv 5888 ctop 20698 TopOnctopon 20715 ct1 21111 cha 21112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topgen 16104 df-top 20699 df-topon 20716 df-cld 20823 df-t1 21118 df-haus 21119 |
This theorem is referenced by: sncld 21175 ishaus3 21626 reghaus 21628 nrmhaus 21629 tgpt1 21921 metreg 22666 ipasslem8 27692 sitmcl 30413 onint1 32448 oninhaus 32449 poimirlem30 33439 |
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