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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 2818 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ispconn 32367 | . 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) |
3 | 2 | simplbi 498 | 1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∪ cuni 4830 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 Topctop 21429 Cn ccn 21760 IIcii 23410 PConncpconn 32363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-pconn 32365 |
This theorem is referenced by: sconntop 32372 pconnconn 32375 txpconn 32376 ptpconn 32377 qtoppconn 32380 pconnpi1 32381 sconnpi1 32383 cvxsconn 32387 |
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