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Mirrors > Home > MPE Home > Th. List > Mathboxes > pconntop | Structured version Visualization version GIF version |
Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
pconntop | ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2826 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ispconn 31752 | . 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) |
3 | 2 | simplbi 493 | 1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 ∃wrex 3119 ∪ cuni 4659 ‘cfv 6124 (class class class)co 6906 0cc0 10253 1c1 10254 Topctop 21069 Cn ccn 21400 IIcii 23049 PConncpconn 31748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-iota 6087 df-fv 6132 df-ov 6909 df-pconn 31750 |
This theorem is referenced by: sconntop 31757 pconnconn 31760 txpconn 31761 ptpconn 31762 qtoppconn 31765 pconnpi1 31766 sconnpi1 31768 cvxsconn 31772 |
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