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Mirrors > Home > MPE Home > Th. List > Mathboxes > onintopssconn | Structured version Visualization version GIF version |
Description: An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.) |
Ref | Expression |
---|---|
onintopssconn | ⊢ (On ∩ Top) ⊆ Conn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3966 | . . 3 ⊢ (𝑥 ∈ (On ∩ Top) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Top)) | |
2 | eloni 6392 | . . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
3 | ordtopconn 36418 | . . . . 5 ⊢ (Ord 𝑥 → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝑥 ∈ On → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn)) |
5 | 4 | biimpa 476 | . . 3 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ Top) → 𝑥 ∈ Conn) |
6 | 1, 5 | sylbi 217 | . 2 ⊢ (𝑥 ∈ (On ∩ Top) → 𝑥 ∈ Conn) |
7 | 6 | ssriv 3986 | 1 ⊢ (On ∩ Top) ⊆ Conn |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∩ cin 3949 ⊆ wss 3950 Ord word 6381 Oncon0 6382 Topctop 22889 Conncconn 23409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-ord 6385 df-on 6386 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-fv 6567 df-topgen 17484 df-top 22890 df-bases 22943 df-cld 23017 df-conn 23410 |
This theorem is referenced by: (None) |
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