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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 35973 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | |
2 | onsucconn 35975 | . . . 4 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Conn) | |
3 | 2 | ordtoplem 35972 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Conn)) |
4 | 1, 3 | sylbid 239 | . 2 ⊢ (Ord 𝐽 → (𝐽 ∈ Top → 𝐽 ∈ Conn)) |
5 | conntop 23334 | . 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 2930 ∪ cuni 4904 Ord word 6364 Topctop 22808 Conncconn 23328 |
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 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-topgen 17419 df-top 22809 df-bases 22862 df-cld 22936 df-conn 23329 |
This theorem is referenced by: onintopssconn 35977 |
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