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Mirrors > Home > MPE Home > Th. List > Mathboxes > sconntop | Structured version Visualization version GIF version |
Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
sconntop | ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sconnpconn 33189 | . 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) | |
2 | pconntop 33187 | . 2 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Topctop 22042 PConncpconn 33181 SConncsconn 33182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-pconn 33183 df-sconn 33184 |
This theorem is referenced by: sconnpi1 33201 txsconn 33203 cvmlift3lem6 33286 cvmlift3lem7 33287 cvmlift3lem8 33288 cvmlift3lem9 33289 |
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