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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 2821 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ispconn 32477 | . 2 ⊢ (𝐽 ∈ PConn ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽∃𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))) |
3 | 2 | simplbi 500 | 1 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ∪ cuni 4824 ‘cfv 6341 (class class class)co 7142 0cc0 10523 1c1 10524 Topctop 21484 Cn ccn 21815 IIcii 23466 PConncpconn 32473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-iota 6300 df-fv 6349 df-ov 7145 df-pconn 32475 |
This theorem is referenced by: sconntop 32482 pconnconn 32485 txpconn 32486 ptpconn 32487 qtoppconn 32490 pconnpi1 32491 sconnpi1 32493 cvxsconn 32497 |
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