![]() |
Mathbox for Chen-Pang He |
< Previous
Next >
Nearby theorems |
|
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 35777 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | |
2 | onsucconn 35779 | . . . 4 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Conn) | |
3 | 2 | ordtoplem 35776 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Conn)) |
4 | 1, 3 | sylbid 239 | . 2 ⊢ (Ord 𝐽 → (𝐽 ∈ Top → 𝐽 ∈ Conn)) |
5 | conntop 23242 | . 2 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | |
6 | 4, 5 | impbid1 224 | 1 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2098 ≠ wne 2932 ∪ cuni 4899 Ord word 6353 Topctop 22716 Conncconn 23236 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-fv 6541 df-topgen 17387 df-top 22717 df-bases 22770 df-cld 22844 df-conn 23237 |
This theorem is referenced by: onintopssconn 35781 |
Copyright terms: Public domain | W3C validator |