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Mirrors > Home > MPE Home > Th. List > is2ndc | Structured version Visualization version GIF version |
Description: The property of being second-countable. (Contributed by Jeff Hankins, 17-Jan-2010.) (Revised by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
is2ndc | ⊢ (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2ndc 21053 | . . 3 ⊢ 2nd𝜔 = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} | |
2 | 1 | eleq2i 2680 | . 2 ⊢ (𝐽 ∈ 2nd𝜔 ↔ 𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}) |
3 | simpr 476 | . . . . 5 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (topGen‘𝑥) = 𝐽) | |
4 | fvex 6113 | . . . . 5 ⊢ (topGen‘𝑥) ∈ V | |
5 | 3, 4 | syl6eqelr 2697 | . . . 4 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) |
6 | 5 | rexlimivw 3011 | . . 3 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) |
7 | eqeq2 2621 | . . . . 5 ⊢ (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽)) | |
8 | 7 | anbi2d 736 | . . . 4 ⊢ (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) |
9 | 8 | rexbidv 3034 | . . 3 ⊢ (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) |
10 | 6, 9 | elab3 3327 | . 2 ⊢ (𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
11 | 2, 10 | bitri 263 | 1 ⊢ (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 ‘cfv 5804 ωcom 6957 ≼ cdom 7839 topGenctg 15921 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-fv 5812 df-2ndc 21053 |
This theorem is referenced by: 2ndctop 21060 2ndci 21061 2ndcsb 21062 2ndcredom 21063 2ndc1stc 21064 2ndcrest 21067 2ndcctbss 21068 2ndcdisj 21069 2ndcomap 21071 2ndcsep 21072 dis2ndc 21073 tx2ndc 21264 |
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