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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ioonct | Structured version Visualization version GIF version |
Description: A nonempty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
ioonct.b | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
ioonct.c | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
ioonct.l | ⊢ (𝜑 → 𝐴 < 𝐵) |
ioonct.a | ⊢ 𝐶 = (𝐴(,)𝐵) |
Ref | Expression |
---|---|
ioonct | ⊢ (𝜑 → ¬ 𝐶 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioontr 40656 | . . . 4 ⊢ ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵) | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = (𝐴(,)𝐵)) |
3 | ioossre 12551 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
4 | 3 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → (𝐴(,)𝐵) ⊆ ℝ) |
5 | ioonct.a | . . . . . . . 8 ⊢ 𝐶 = (𝐴(,)𝐵) | |
6 | 5 | breq1i 4895 | . . . . . . 7 ⊢ (𝐶 ≼ ω ↔ (𝐴(,)𝐵) ≼ ω) |
7 | 6 | biimpi 208 | . . . . . 6 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ω) |
8 | nnenom 13102 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
9 | 8 | ensymi 8293 | . . . . . . 7 ⊢ ω ≈ ℕ |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝐶 ≼ ω → ω ≈ ℕ) |
11 | domentr 8302 | . . . . . 6 ⊢ (((𝐴(,)𝐵) ≼ ω ∧ ω ≈ ℕ) → (𝐴(,)𝐵) ≼ ℕ) | |
12 | 7, 10, 11 | syl2anc 579 | . . . . 5 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ℕ) |
13 | 12 | adantl 475 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → (𝐴(,)𝐵) ≼ ℕ) |
14 | rectbntr0 23047 | . . . 4 ⊢ (((𝐴(,)𝐵) ⊆ ℝ ∧ (𝐴(,)𝐵) ≼ ℕ) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) | |
15 | 4, 13, 14 | syl2anc 579 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) |
16 | 2, 15 | eqtr3d 2816 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → (𝐴(,)𝐵) = ∅) |
17 | ioonct.l | . . . . 5 ⊢ (𝜑 → 𝐴 < 𝐵) | |
18 | ioonct.b | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
19 | ioonct.c | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
20 | ioon0 12517 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) | |
21 | 18, 19, 20 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) |
22 | 17, 21 | mpbird 249 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅) |
23 | 22 | neneqd 2974 | . . 3 ⊢ (𝜑 → ¬ (𝐴(,)𝐵) = ∅) |
24 | 23 | adantr 474 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ¬ (𝐴(,)𝐵) = ∅) |
25 | 16, 24 | pm2.65da 807 | 1 ⊢ (𝜑 → ¬ 𝐶 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ⊆ wss 3792 ∅c0 4141 class class class wbr 4888 ran crn 5358 ‘cfv 6137 (class class class)co 6924 ωcom 7345 ≈ cen 8240 ≼ cdom 8241 ℝcr 10273 ℝ*cxr 10412 < clt 10413 ℕcn 11378 (,)cioo 12491 topGenctg 16488 intcnt 21233 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-omul 7850 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-acn 9103 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-n0 11647 df-z 11733 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-fl 12916 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-limsup 14614 df-clim 14631 df-rlim 14632 df-sum 14829 df-topgen 16494 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-top 21110 df-topon 21127 df-bases 21162 df-ntr 21236 |
This theorem is referenced by: iocnct 40685 iccnct 40686 |
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