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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 22045 | . . 3 ⊢ 2ndω = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} | |
2 | 1 | eleq2i 2881 | . 2 ⊢ (𝐽 ∈ 2ndω ↔ 𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)}) |
3 | simpr 488 | . . . . 5 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (topGen‘𝑥) = 𝐽) | |
4 | fvex 6658 | . . . . 5 ⊢ (topGen‘𝑥) ∈ V | |
5 | 3, 4 | eqeltrrdi 2899 | . . . 4 ⊢ ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) |
6 | 5 | rexlimivw 3241 | . . 3 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ∈ V) |
7 | eqeq2 2810 | . . . . 5 ⊢ (𝑗 = 𝐽 → ((topGen‘𝑥) = 𝑗 ↔ (topGen‘𝑥) = 𝐽)) | |
8 | 7 | anbi2d 631 | . . . 4 ⊢ (𝑗 = 𝐽 → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) |
9 | 8 | rexbidv 3256 | . . 3 ⊢ (𝑗 = 𝐽 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗) ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))) |
10 | 6, 9 | elab3 3622 | . 2 ⊢ (𝐽 ∈ {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
11 | 2, 10 | bitri 278 | 1 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 ωcom 7560 ≼ cdom 8490 topGenctg 16703 TopBasesctb 21550 2ndωc2ndc 22043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-uni 4801 df-iota 6283 df-fv 6332 df-2ndc 22045 |
This theorem is referenced by: 2ndctop 22052 2ndci 22053 2ndcsb 22054 2ndcredom 22055 2ndc1stc 22056 2ndcrest 22059 2ndcctbss 22060 2ndcdisj 22061 2ndcomap 22063 2ndcsep 22064 dis2ndc 22065 tx2ndc 22256 |
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