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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 3960 | . . 3 ⊢ (𝑥 ∈ (On ∩ Top) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Top)) | |
2 | eloni 6373 | . . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
3 | ordtopconn 35859 | . . . . 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 216 | . 2 ⊢ (𝑥 ∈ (On ∩ Top) → 𝑥 ∈ Conn) |
7 | 6 | ssriv 3982 | 1 ⊢ (On ∩ Top) ⊆ Conn |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ∩ cin 3943 ⊆ wss 3944 Ord word 6362 Oncon0 6363 Topctop 22782 Conncconn 23302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 df-topgen 17416 df-top 22783 df-bases 22836 df-cld 22910 df-conn 23303 |
This theorem is referenced by: (None) |
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