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Mirrors > Home > MPE Home > Th. List > conntop | Structured version Visualization version GIF version |
Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.) |
Ref | Expression |
---|---|
conntop | ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | isconn 21216 | . 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽})) |
3 | 2 | simplbi 476 | 1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ∅c0 3915 {cpr 4179 ∪ cuni 4436 ‘cfv 5888 Topctop 20698 Clsdccld 20820 Conncconn 21214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-conn 21215 |
This theorem is referenced by: conncompss 21236 txconn 21492 qtopconn 21512 ufildr 21735 connpconn 31217 cvmliftmolem1 31263 cvmliftmolem2 31264 ordtopconn 32438 |
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