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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsuctop | Structured version Visualization version GIF version |
Description: A successor ordinal number is a topology. (Contributed by Chen-Pang He, 11-Oct-2015.) |
Ref | Expression |
---|---|
onsuctop | β’ (π΄ β On β suc π΄ β Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ontgsucval 35620 | . 2 β’ (π΄ β On β (topGenβsuc π΄) = suc π΄) | |
2 | onsuc 7801 | . . 3 β’ (π΄ β On β suc π΄ β On) | |
3 | ontopbas 35616 | . . 3 β’ (suc π΄ β On β suc π΄ β TopBases) | |
4 | tgcl 22692 | . . 3 β’ (suc π΄ β TopBases β (topGenβsuc π΄) β Top) | |
5 | 2, 3, 4 | 3syl 18 | . 2 β’ (π΄ β On β (topGenβsuc π΄) β Top) |
6 | 1, 5 | eqeltrrd 2832 | 1 β’ (π΄ β On β suc π΄ β Top) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2104 Oncon0 6363 suc csuc 6365 βcfv 6542 topGenctg 17387 Topctop 22615 TopBasesctb 22668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fv 6550 df-topgen 17393 df-top 22616 df-bases 22669 |
This theorem is referenced by: onsuctopon 35622 ordtop 35624 onsucconni 35625 onsucsuccmpi 35631 |
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