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Mirrors > Home > MPE Home > Th. List > metelcls | Structured version Visualization version GIF version |
Description: A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 9572. The statement can be generalized to first-countable spaces, not just metrizable spaces. (Contributed by NM, 8-Nov-2007.) (Proof shortened by Mario Carneiro, 1-May-2015.) |
Ref | Expression |
---|---|
metelcls.2 | ⊢ 𝐽 = (MetOpen‘𝐷) |
metelcls.3 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
metelcls.5 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
Ref | Expression |
---|---|
metelcls | ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metelcls.3 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
2 | metelcls.2 | . . . 4 ⊢ 𝐽 = (MetOpen‘𝐷) | |
3 | 2 | met1stc 22696 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1st𝜔) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝐽 ∈ 1st𝜔) |
5 | metelcls.5 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
6 | 2 | mopnuni 22616 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
7 | 1, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
8 | 5, 7 | sseqtrd 3866 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽) |
9 | eqid 2825 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 9 | 1stcelcls 21635 | . 2 ⊢ ((𝐽 ∈ 1st𝜔 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) |
11 | 4, 8, 10 | syl2anc 581 | 1 ⊢ (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆 ∧ 𝑓(⇝𝑡‘𝐽)𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ⊆ wss 3798 ∪ cuni 4658 class class class wbr 4873 ⟶wf 6119 ‘cfv 6123 ℕcn 11350 ∞Metcxmet 20091 MetOpencmopn 20096 clsccl 21193 ⇝𝑡clm 21401 1st𝜔c1stc 21611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cc 9572 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-card 9078 df-acn 9081 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-fz 12620 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-bl 20101 df-mopn 20102 df-top 21069 df-topon 21086 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-lm 21404 df-1stc 21613 |
This theorem is referenced by: metcld 23474 |
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