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Mirrors > Home > MPE Home > Th. List > nrmtop | Structured version Visualization version GIF version |
Description: A normal space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
nrmtop | ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnrm 22088 | . 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝐽 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐽 ∈ Nrm → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 ∩ cin 3842 ⊆ wss 3843 𝒫 cpw 4488 ‘cfv 6339 Topctop 21646 Clsdccld 21769 clsccl 21771 Nrmcnrm 22063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-iota 6297 df-fv 6347 df-nrm 22070 |
This theorem is referenced by: pnrmtop 22094 nrmsep 22110 isnrm2 22111 isnrm3 22112 nrmr0reg 22502 kqnrm 22505 nrmhmph 22547 |
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