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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 32772 | . 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) | |
2 | pconntop 32770 | . 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 2114 Topctop 21656 PConncpconn 32764 SConncsconn 32765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-un 3858 df-in 3860 df-ss 3870 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6307 df-fv 6357 df-ov 7185 df-pconn 32766 df-sconn 32767 |
This theorem is referenced by: sconnpi1 32784 txsconn 32786 cvmlift3lem6 32869 cvmlift3lem7 32870 cvmlift3lem8 32871 cvmlift3lem9 32872 |
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