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Mirrors > Home > MPE Home > Th. List > 2ndctop | Structured version Visualization version GIF version |
Description: A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
2ndctop | β’ (π½ β 2ndΟ β π½ β Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | is2ndc 22820 | . 2 β’ (π½ β 2ndΟ β βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) | |
2 | simprr 772 | . . . 4 β’ ((π₯ β TopBases β§ (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) β (topGenβπ₯) = π½) | |
3 | tgcl 22342 | . . . . 5 β’ (π₯ β TopBases β (topGenβπ₯) β Top) | |
4 | 3 | adantr 482 | . . . 4 β’ ((π₯ β TopBases β§ (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) β (topGenβπ₯) β Top) |
5 | 2, 4 | eqeltrrd 2835 | . . 3 β’ ((π₯ β TopBases β§ (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) β π½ β Top) |
6 | 5 | rexlimiva 3141 | . 2 β’ (βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½) β π½ β Top) |
7 | 1, 6 | sylbi 216 | 1 β’ (π½ β 2ndΟ β π½ β Top) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3070 class class class wbr 5109 βcfv 6500 Οcom 7806 βΌ cdom 8887 topGenctg 17327 Topctop 22265 TopBasesctb 22318 2ndΟc2ndc 22812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-iota 6452 df-fun 6502 df-fv 6508 df-topgen 17333 df-top 22266 df-bases 22319 df-2ndc 22814 |
This theorem is referenced by: 2ndc1stc 22825 2ndcctbss 22829 |
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