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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 23264 | . . 3 ⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} | |
2 | 1 | eleq2i 2824 | . 2 ⊢ (𝐽 ∈ 2ndω ↔ 𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}) |
3 | simpr 484 | . . . . 5 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (topGen‘𝑥) = 𝐽) | |
4 | fvex 6904 | . . . . 5 ⊢ (topGen‘𝑥) ∈ V | |
5 | 3, 4 | eqeltrrdi 2841 | . . . 4 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) |
6 | 5 | rexlimivw 3150 | . . 3 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) |
7 | eqeq2 2743 | . . . . 5 ⊢ (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽)) | |
8 | 7 | anbi2d 628 | . . . 4 ⊢ (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) |
9 | 8 | rexbidv 3177 | . . 3 ⊢ (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) |
10 | 6, 9 | elab3 3676 | . 2 ⊢ (𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
11 | 2, 10 | bitri 275 | 1 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {cab 2708 ∃wrex 3069 Vcvv 3473 class class class wbr 5148 ‘cfv 6543 ωcom 7859 ≼ cdom 8943 topGenctg 17390 TopBasesctb 22768 2ndωc2ndc 23262 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-rex 3070 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-sn 4629 df-pr 4631 df-uni 4909 df-iota 6495 df-fv 6551 df-2ndc 23264 |
This theorem is referenced by: 2ndctop 23271 2ndci 23272 2ndcsb 23273 2ndcredom 23274 2ndc1stc 23275 2ndcrest 23278 2ndcctbss 23279 2ndcdisj 23280 2ndcomap 23282 2ndcsep 23283 dis2ndc 23284 tx2ndc 23475 |
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