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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 | ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ontgsucval 33893 | . 2 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) | |
2 | suceloni 7508 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
3 | ontopbas 33889 | . . 3 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ TopBases) | |
4 | tgcl 21574 | . . 3 ⊢ (suc 𝐴 ∈ TopBases → (topGen‘suc 𝐴) ∈ Top) | |
5 | 2, 3, 4 | 3syl 18 | . 2 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) ∈ Top) |
6 | 1, 5 | eqeltrrd 2891 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Oncon0 6159 suc csuc 6161 ‘cfv 6324 topGenctg 16703 Topctop 21498 TopBasesctb 21550 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fv 6332 df-topgen 16709 df-top 21499 df-bases 21551 |
This theorem is referenced by: onsuctopon 33895 ordtop 33897 onsucconni 33898 onsucsuccmpi 33904 |
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