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Mirrors > Home > MPE Home > Th. List > is2ndc | Structured version Visualization version GIF version |
Description: The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
is2ndc | β’ (π½ β 2ndΟ β βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2ndc 23357 | . . 3 β’ 2ndΟ = {π β£ βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π)} | |
2 | 1 | eleq2i 2821 | . 2 β’ (π½ β 2ndΟ β π½ β {π β£ βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π)}) |
3 | simpr 484 | . . . . 5 β’ ((π₯ βΌ Ο β§ (topGenβπ₯) = π½) β (topGenβπ₯) = π½) | |
4 | fvex 6910 | . . . . 5 β’ (topGenβπ₯) β V | |
5 | 3, 4 | eqeltrrdi 2838 | . . . 4 β’ ((π₯ βΌ Ο β§ (topGenβπ₯) = π½) β π½ β V) |
6 | 5 | rexlimivw 3148 | . . 3 β’ (βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½) β π½ β V) |
7 | eqeq2 2740 | . . . . 5 β’ (π = π½ β ((topGenβπ₯) = π β (topGenβπ₯) = π½)) | |
8 | 7 | anbi2d 629 | . . . 4 β’ (π = π½ β ((π₯ βΌ Ο β§ (topGenβπ₯) = π) β (π₯ βΌ Ο β§ (topGenβπ₯) = π½))) |
9 | 8 | rexbidv 3175 | . . 3 β’ (π = π½ β (βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π) β βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½))) |
10 | 6, 9 | elab3 3675 | . 2 β’ (π½ β {π β£ βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π)} β βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) |
11 | 2, 10 | bitri 275 | 1 β’ (π½ β 2ndΟ β βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 {cab 2705 βwrex 3067 Vcvv 3471 class class class wbr 5148 βcfv 6548 Οcom 7870 βΌ cdom 8962 topGenctg 17419 TopBasesctb 22861 2ndΟc2ndc 23355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rex 3068 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-sn 4630 df-pr 4632 df-uni 4909 df-iota 6500 df-fv 6556 df-2ndc 23357 |
This theorem is referenced by: 2ndctop 23364 2ndci 23365 2ndcsb 23366 2ndcredom 23367 2ndc1stc 23368 2ndcrest 23371 2ndcctbss 23372 2ndcdisj 23373 2ndcomap 23375 2ndcsep 23376 dis2ndc 23377 tx2ndc 23568 |
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