|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 23449 | . . 3 ⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} | |
| 2 | 1 | eleq2i 2832 | . 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 6918 | . . . . 5 ⊢ (topGen‘𝑥) ∈ V | |
| 5 | 3, 4 | eqeltrrdi 2849 | . . . 4 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) | 
| 6 | 5 | rexlimivw 3150 | . . 3 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) | 
| 7 | eqeq2 2748 | . . . . 5 ⊢ (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽)) | |
| 8 | 7 | anbi2d 630 | . . . 4 ⊢ (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) | 
| 9 | 8 | rexbidv 3178 | . . 3 ⊢ (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) | 
| 10 | 6, 9 | elab3 3685 | . 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 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 Vcvv 3479 class class class wbr 5142 ‘cfv 6560 ωcom 7888 ≼ cdom 8984 topGenctg 17483 TopBasesctb 22953 2ndωc2ndc 23447 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rex 3070 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-sn 4626 df-pr 4628 df-uni 4907 df-iota 6513 df-fv 6568 df-2ndc 23449 | 
| This theorem is referenced by: 2ndctop 23456 2ndci 23457 2ndcsb 23458 2ndcredom 23459 2ndc1stc 23460 2ndcrest 23463 2ndcctbss 23464 2ndcdisj 23465 2ndcomap 23467 2ndcsep 23468 dis2ndc 23469 tx2ndc 23660 | 
| Copyright terms: Public domain | W3C validator |