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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 35305 | . 2 ⊢ (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴) | |
2 | onsuc 7795 | . . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On) | |
3 | ontopbas 35301 | . . 3 ⊢ (suc 𝐴 ∈ On → suc 𝐴 ∈ TopBases) | |
4 | tgcl 22463 | . . 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 2834 | 1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Oncon0 6361 suc csuc 6363 ‘cfv 6540 topGenctg 17379 Topctop 22386 TopBasesctb 22439 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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 6364 df-on 6365 df-suc 6367 df-iota 6492 df-fun 6542 df-fv 6548 df-topgen 17385 df-top 22387 df-bases 22440 |
This theorem is referenced by: onsuctopon 35307 ordtop 35309 onsucconni 35310 onsucsuccmpi 35316 |
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