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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 2771 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ispconn 31543 | . 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) |
3 | 2 | simplbi 485 | 1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 ∪ cuni 4574 ‘cfv 6031 (class class class)co 6793 0cc0 10138 1c1 10139 Topctop 20918 Cn ccn 21249 IIcii 22898 PConncpconn 31539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-iota 5994 df-fv 6039 df-ov 6796 df-pconn 31541 |
This theorem is referenced by: sconntop 31548 pconnconn 31551 txpconn 31552 ptpconn 31553 qtoppconn 31556 pconnpi1 31557 sconnpi1 31559 cvxsconn 31563 |
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