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Mirrors > Home > MPE Home > Th. List > haust1 | Structured 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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2451 |
. . . . . . . . 9
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2 | 1 | hausnei 19048 |
. . . . . . . 8
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3 | simprr1 1036 |
. . . . . . . . . . 11
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4 | noel 3739 |
. . . . . . . . . . . . 13
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5 | simprr3 1038 |
. . . . . . . . . . . . . 14
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6 | 5 | eleq2d 2521 |
. . . . . . . . . . . . 13
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7 | 4, 6 | mtbiri 303 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | simprr2 1037 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | elin 3637 |
. . . . . . . . . . . . . 14
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10 | 9 | simplbi2com 627 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 8, 10 | syl 16 |
. . . . . . . . . . . 12
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12 | 7, 11 | mtod 177 |
. . . . . . . . . . 11
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13 | 3, 12 | jca 532 |
. . . . . . . . . 10
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14 | 13 | rexlimdvaa 2938 |
. . . . . . . . 9
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15 | 14 | reximdva 2924 |
. . . . . . . 8
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16 | 2, 15 | mpd 15 |
. . . . . . 7
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17 | rexanali 2870 |
. . . . . . 7
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18 | 16, 17 | sylib 196 |
. . . . . 6
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19 | 18 | 3exp2 1206 |
. . . . 5
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20 | 19 | imp32 433 |
. . . 4
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21 | 20 | necon4ad 2668 |
. . 3
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22 | 21 | ralrimivva 2904 |
. 2
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23 | haustop 19051 |
. . . 4
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24 | 1 | toptopon 18654 |
. . . 4
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25 | 23, 24 | sylib 196 |
. . 3
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26 | ist1-2 19067 |
. . 3
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27 | 25, 26 | syl 16 |
. 2
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28 | 22, 27 | mpbird 232 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-rab 2804 df-v 3070 df-sbc 3285 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-op 3982 df-uni 4190 df-br 4391 df-opab 4449 df-mpt 4450 df-id 4734 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-iota 5479 df-fun 5518 df-fv 5524 df-topgen 14484 df-top 18619 df-topon 18622 df-cld 18739 df-t1 19034 df-haus 19035 |
This theorem is referenced by: sncld 19091 ishaus3 19512 reghaus 19514 nrmhaus 19515 tgpt1 19804 metreg 20555 ipasslem8 24372 onint1 28429 oninhaus 28430 |
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