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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 33779 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ≠ ∪ 𝐽)) | |
2 | onsucconn 33781 | . . . 4 ⊢ (∪ 𝐽 ∈ On → suc ∪ 𝐽 ∈ Conn) | |
3 | 2 | ordtoplem 33778 | . . 3 ⊢ (Ord 𝐽 → (𝐽 ≠ ∪ 𝐽 → 𝐽 ∈ Conn)) |
4 | 1, 3 | sylbid 242 | . 2 ⊢ (Ord 𝐽 → (𝐽 ∈ Top → 𝐽 ∈ Conn)) |
5 | conntop 22019 | . 2 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | |
6 | 4, 5 | impbid1 227 | 1 ⊢ (Ord 𝐽 → (𝐽 ∈ Top ↔ 𝐽 ∈ Conn)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2110 ≠ wne 3016 ∪ cuni 4832 Ord word 6185 Topctop 21495 Conncconn 22013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-ord 6189 df-on 6190 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-fv 6358 df-topgen 16711 df-top 21496 df-bases 21548 df-cld 21621 df-conn 22014 |
This theorem is referenced by: onintopssconn 33783 |
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