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Mirrors > Home > MPE Home > Th. List > uspreg | Structured version Unicode version |
Description: If a uniform space is Hausdorff, it is regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 4-Jan-2018.) |
Ref | Expression |
---|---|
uspreg.1 |
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Ref | Expression |
---|---|
uspreg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2452 |
. . . . 5
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2 | eqid 2452 |
. . . . 5
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3 | uspreg.1 |
. . . . 5
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4 | 1, 2, 3 | isusp 19963 |
. . . 4
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5 | 4 | simprbi 464 |
. . 3
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6 | 5 | adantr 465 |
. 2
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7 | 4 | simplbi 460 |
. . . 4
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8 | 7 | adantr 465 |
. . 3
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9 | simpr 461 |
. . . 4
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10 | 6, 9 | eqeltrrd 2541 |
. . 3
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11 | eqid 2452 |
. . . 4
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12 | 11 | utopreg 19954 |
. . 3
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13 | 8, 10, 12 | syl2anc 661 |
. 2
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14 | 6, 13 | eqeltrd 2540 |
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 1954 ax-ext 2431 ax-rep 4506 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-reu 2803 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-pss 3447 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4195 df-int 4232 df-iun 4276 df-iin 4277 df-br 4396 df-opab 4454 df-mpt 4455 df-tr 4489 df-eprel 4735 df-id 4739 df-po 4744 df-so 4745 df-fr 4782 df-we 4784 df-ord 4825 df-on 4826 df-lim 4827 df-suc 4828 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fn 5524 df-f 5525 df-f1 5526 df-fo 5527 df-f1o 5528 df-fv 5529 df-ov 6198 df-oprab 6199 df-mpt2 6200 df-om 6582 df-1st 6682 df-2nd 6683 df-recs 6937 df-rdg 6971 df-1o 7025 df-oadd 7029 df-er 7206 df-map 7321 df-en 7416 df-fin 7419 df-fi 7767 df-topgen 14496 df-top 18630 df-bases 18632 df-topon 18633 df-cld 18750 df-ntr 18751 df-cls 18752 df-nei 18829 df-cn 18958 df-cnp 18959 df-reg 19047 df-tx 19262 df-ust 19902 df-utop 19933 df-usp 19959 |
This theorem is referenced by: cnextucn 20005 rrhre 26587 |
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