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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 22791 | . . 3 β’ 2ndΟ = {π β£ βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π)} | |
2 | 1 | eleq2i 2829 | . 2 β’ (π½ β 2ndΟ β π½ β {π β£ βπ₯ β TopBases (π₯ βΌ Ο β§ (topGenβπ₯) = π)}) |
3 | simpr 485 | . . . . 5 β’ ((π₯ βΌ Ο β§ (topGenβπ₯) = π½) β (topGenβπ₯) = π½) | |
4 | fvex 6855 | . . . . 5 β’ (topGenβπ₯) β V | |
5 | 3, 4 | eqeltrrdi 2846 | . . . 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 2748 | . . . . 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 3638 | . 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 274 | 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 396 = wceq 1541 β wcel 2106 {cab 2713 βwrex 3073 Vcvv 3445 class class class wbr 5105 βcfv 6496 Οcom 7802 βΌ cdom 8881 topGenctg 17319 TopBasesctb 22295 2ndΟc2ndc 22789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5263 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-rex 3074 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-sn 4587 df-pr 4589 df-uni 4866 df-iota 6448 df-fv 6504 df-2ndc 22791 |
This theorem is referenced by: 2ndctop 22798 2ndci 22799 2ndcsb 22800 2ndcredom 22801 2ndc1stc 22802 2ndcrest 22805 2ndcctbss 22806 2ndcdisj 22807 2ndcomap 22809 2ndcsep 22810 dis2ndc 22811 tx2ndc 23002 |
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