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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 32084 | . 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) | |
2 | pconntop 32082 | . 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 2083 Topctop 21189 PConncpconn 32076 SConncsconn 32077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-iota 6196 df-fv 6240 df-ov 7026 df-pconn 32078 df-sconn 32079 |
This theorem is referenced by: sconnpi1 32096 txsconn 32098 cvmlift3lem6 32181 cvmlift3lem7 32182 cvmlift3lem8 32183 cvmlift3lem9 32184 |
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