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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordtopt0 | Structured version Visualization version GIF version |
Description: An ordinal topology is T0. (Contributed by Chen-Pang He, 8-Nov-2015.) |
Ref | Expression |
---|---|
ordtopt0 | ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtop 35829 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | |
2 | onsuct0 35834 | . . . 4 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Kol2) | |
3 | 2 | ordtoplem 35828 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Kol2)) |
4 | 1, 3 | sylbid 239 | . 2 ⊢ (Ord 𝐽 → (𝐽 ∈ Top → 𝐽 ∈ Kol2)) |
5 | t0top 23188 | . 2 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | |
6 | 4, 5 | impbid1 224 | 1 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Kol2)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 ≠ wne 2934 ∪ cuni 4902 Ord word 6357 Topctop 22750 Kol2ct0 23165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-ord 6361 df-on 6362 df-suc 6364 df-iota 6489 df-fun 6539 df-fv 6545 df-topgen 17398 df-top 22751 df-topon 22768 df-bases 22804 df-t0 23172 |
This theorem is referenced by: (None) |
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