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Mirrors > Home > MPE Home > Th. List > 2ndctop | Structured version Visualization version GIF version |
Description: A second-countable topology is a topology. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
2ndctop | ⊢ (𝐽 ∈ 2nd𝜔 → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | is2ndc 21059 | . 2 ⊢ (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) | |
2 | simprr 792 | . . . 4 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) = 𝐽) | |
3 | tgcl 20584 | . . . . 5 ⊢ (𝑥 ∈ TopBases → (topGen‘𝑥) ∈ Top) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → (topGen‘𝑥) ∈ Top) |
5 | 2, 4 | eqeltrrd 2689 | . . 3 ⊢ ((𝑥 ∈ TopBases ∧ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) → 𝐽 ∈ Top) |
6 | 5 | rexlimiva 3010 | . 2 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ Top) |
7 | 1, 6 | sylbi 206 | 1 ⊢ (𝐽 ∈ 2nd𝜔 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 ωcom 6957 ≼ cdom 7839 topGenctg 15921 Topctop 20517 TopBasesctb 20520 2nd𝜔c2ndc 21051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-topgen 15927 df-top 20521 df-bases 20522 df-2ndc 21053 |
This theorem is referenced by: 2ndc1stc 21064 2ndcctbss 21068 |
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