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Mirrors > Home > MPE Home > Th. List > nrmhaus | Structured version Visualization version Unicode version |
Description: A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
nrmhaus |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haust1 20380 |
. 2
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2 | nrmreg 20851 |
. . . 4
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3 | 2 | ex 436 |
. . 3
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4 | t1t0 20376 |
. . . 4
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5 | reghaus 20852 |
. . . 4
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6 | 4, 5 | syl5ibrcom 226 |
. . 3
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7 | 3, 6 | sylcom 30 |
. 2
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8 | 1, 7 | impbid2 208 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-iin 4284 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-1st 6798 df-2nd 6799 df-1o 7187 df-map 7479 df-topgen 15354 df-qtop 15418 df-top 19933 df-topon 19935 df-cld 20046 df-cls 20048 df-cn 20255 df-t0 20341 df-t1 20342 df-haus 20343 df-reg 20344 df-nrm 20345 df-kq 20721 df-hmeo 20782 df-hmph 20783 |
This theorem is referenced by: (None) |
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