Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 2ndcredom | Structured version Visualization version GIF version |
Description: A second-countable space has at most the cardinality of the continuum. (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
2ndcredom | ⊢ (𝐽 ∈ 2ndω → 𝐽 ≼ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | is2ndc 21982 | . 2 ⊢ (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽)) | |
2 | tgdom 21514 | . . . . . 6 ⊢ (𝑥 ∈ TopBases → (topGen‘𝑥) ≼ 𝒫 𝑥) | |
3 | simpr 485 | . . . . . . . . 9 ⊢ ((𝑥 ∈ TopBases ∧ 𝑥 ≼ ω) → 𝑥 ≼ ω) | |
4 | nnenom 13336 | . . . . . . . . . 10 ⊢ ℕ ≈ ω | |
5 | 4 | ensymi 8547 | . . . . . . . . 9 ⊢ ω ≈ ℕ |
6 | domentr 8556 | . . . . . . . . 9 ⊢ ((𝑥 ≼ ω ∧ ω ≈ ℕ) → 𝑥 ≼ ℕ) | |
7 | 3, 5, 6 | sylancl 586 | . . . . . . . 8 ⊢ ((𝑥 ∈ TopBases ∧ 𝑥 ≼ ω) → 𝑥 ≼ ℕ) |
8 | pwdom 8657 | . . . . . . . 8 ⊢ (𝑥 ≼ ℕ → 𝒫 𝑥 ≼ 𝒫 ℕ) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ ((𝑥 ∈ TopBases ∧ 𝑥 ≼ ω) → 𝒫 𝑥 ≼ 𝒫 ℕ) |
10 | rpnnen 15568 | . . . . . . . 8 ⊢ ℝ ≈ 𝒫 ℕ | |
11 | 10 | ensymi 8547 | . . . . . . 7 ⊢ 𝒫 ℕ ≈ ℝ |
12 | domentr 8556 | . . . . . . 7 ⊢ ((𝒫 𝑥 ≼ 𝒫 ℕ ∧ 𝒫 ℕ ≈ ℝ) → 𝒫 𝑥 ≼ ℝ) | |
13 | 9, 11, 12 | sylancl 586 | . . . . . 6 ⊢ ((𝑥 ∈ TopBases ∧ 𝑥 ≼ ω) → 𝒫 𝑥 ≼ ℝ) |
14 | domtr 8550 | . . . . . 6 ⊢ (((topGen‘𝑥) ≼ 𝒫 𝑥 ∧ 𝒫 𝑥 ≼ ℝ) → (topGen‘𝑥) ≼ ℝ) | |
15 | 2, 13, 14 | syl2an2r 681 | . . . . 5 ⊢ ((𝑥 ∈ TopBases ∧ 𝑥 ≼ ω) → (topGen‘𝑥) ≼ ℝ) |
16 | breq1 5060 | . . . . 5 ⊢ ((topGen‘𝑥) = 𝐽 → ((topGen‘𝑥) ≼ ℝ ↔ 𝐽 ≼ ℝ)) | |
17 | 15, 16 | syl5ibcom 246 | . . . 4 ⊢ ((𝑥 ∈ TopBases ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) = 𝐽 → 𝐽 ≼ ℝ)) |
18 | 17 | expimpd 454 | . . 3 ⊢ (𝑥 ∈ TopBases → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ≼ ℝ)) |
19 | 18 | rexlimiv 3277 | . 2 ⊢ (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → 𝐽 ≼ ℝ) |
20 | 1, 19 | sylbi 218 | 1 ⊢ (𝐽 ∈ 2ndω → 𝐽 ≼ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 𝒫 cpw 4535 class class class wbr 5057 ‘cfv 6348 ωcom 7569 ≈ cen 8494 ≼ cdom 8495 ℝcr 10524 ℕcn 11626 topGenctg 16699 TopBasesctb 21481 2ndωc2ndc 21974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-topgen 16705 df-2ndc 21976 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |