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Mirrors > Home > MPE Home > Th. List > retopconn | Structured version Visualization version GIF version |
Description: Corollary of reconn 23972. The set of real numbers is connected. (Contributed by Jeff Hankins, 17-Aug-2009.) |
Ref | Expression |
---|---|
retopconn | ⊢ (topGen‘ran (,)) ∈ Conn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retop 23906 | . . 3 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | uniretop 23907 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
3 | 2 | restid 17125 | . . 3 ⊢ ((topGen‘ran (,)) ∈ Top → ((topGen‘ran (,)) ↾t ℝ) = (topGen‘ran (,))) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ ((topGen‘ran (,)) ↾t ℝ) = (topGen‘ran (,)) |
5 | iccssre 13143 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥[,]𝑦) ⊆ ℝ) | |
6 | 5 | rgen2 3128 | . . 3 ⊢ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥[,]𝑦) ⊆ ℝ |
7 | ssid 3947 | . . . 4 ⊢ ℝ ⊆ ℝ | |
8 | reconn 23972 | . . . 4 ⊢ (ℝ ⊆ ℝ → (((topGen‘ran (,)) ↾t ℝ) ∈ Conn ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥[,]𝑦) ⊆ ℝ)) | |
9 | 7, 8 | ax-mp 5 | . . 3 ⊢ (((topGen‘ran (,)) ↾t ℝ) ∈ Conn ↔ ∀𝑥 ∈ ℝ ∀𝑦 ∈ ℝ (𝑥[,]𝑦) ⊆ ℝ) |
10 | 6, 9 | mpbir 230 | . 2 ⊢ ((topGen‘ran (,)) ↾t ℝ) ∈ Conn |
11 | 4, 10 | eqeltrri 2837 | 1 ⊢ (topGen‘ran (,)) ∈ Conn |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2109 ∀wral 3065 ⊆ wss 3891 ran crn 5589 ‘cfv 6430 (class class class)co 7268 ℝcr 10854 (,)cioo 13061 [,]cicc 13064 ↾t crest 17112 topGenctg 17129 Topctop 22023 Conncconn 22543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fi 9131 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-z 12303 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-ico 13067 df-icc 13068 df-seq 13703 df-exp 13764 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-rest 17114 df-topgen 17135 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-top 22024 df-topon 22041 df-bases 22077 df-cld 22151 df-conn 22544 |
This theorem is referenced by: mblfinlem1 35793 |
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