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Mirrors > Home > MPE Home > Th. List > Mathboxes > ordtopconn | Structured version Visualization version GIF version |
Description: An ordinal topology is connected. (Contributed by Chen-Pang He, 1-Nov-2015.) |
Ref | Expression |
---|---|
ordtopconn | ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtop 34319 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | |
2 | onsucconn 34321 | . . . 4 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Conn) | |
3 | 2 | ordtoplem 34318 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Conn)) |
4 | 1, 3 | sylbid 243 | . 2 ⊢ (Ord 𝐽 → (𝐽 ∈ Top → 𝐽 ∈ Conn)) |
5 | conntop 22286 | . 2 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | |
6 | 4, 5 | impbid1 228 | 1 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2110 ≠ wne 2935 ∪ cuni 4809 Ord word 6201 Topctop 21762 Conncconn 22280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-ord 6205 df-on 6206 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-fv 6377 df-topgen 16920 df-top 21763 df-bases 21815 df-cld 21888 df-conn 22281 |
This theorem is referenced by: onintopssconn 34323 |
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