Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ioonct | Structured version Visualization version GIF version |
Description: C non empty 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 39736 | . . . 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 12235 | . . . . 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 4660 | . . . . . . 7 ⊢ (𝐶 ≼ ω ↔ (𝐴(,)𝐵) ≼ ω) |
7 | 6 | biimpi 206 | . . . . . 6 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ω) |
8 | nnenom 12779 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
9 | 8 | ensymi 8006 | . . . . . . 7 ⊢ ω ≈ ℕ |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝐶 ≼ ω → ω ≈ ℕ) |
11 | domentr 8015 | . . . . . 6 ⊢ (((𝐴(,)𝐵) ≼ ω ∧ ω ≈ ℕ) → (𝐴(,)𝐵) ≼ ℕ) | |
12 | 7, 10, 11 | syl2anc 693 | . . . . 5 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ℕ) |
13 | 12 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → (𝐴(,)𝐵) ≼ ℕ) |
14 | rectbntr0 22635 | . . . 4 ⊢ (((𝐴(,)𝐵) ⊆ ℝ ∧ (𝐴(,)𝐵) ≼ ℕ) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) | |
15 | 4, 13, 14 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) |
16 | 2, 15 | eqtr3d 2658 | . 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 12201 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) | |
21 | 18, 19, 20 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) |
22 | 17, 21 | mpbird 247 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅) |
23 | 22 | neneqd 2799 | . . 3 ⊢ (𝜑 → ¬ (𝐴(,)𝐵) = ∅) |
24 | 23 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ¬ (𝐴(,)𝐵) = ∅) |
25 | 16, 24 | pm2.65da 600 | 1 ⊢ (𝜑 → ¬ 𝐶 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ωcom 7065 ≈ cen 7952 ≼ cdom 7953 ℝcr 9935 ℝ*cxr 10073 < clt 10074 ℕcn 11020 (,)cioo 12175 topGenctg 16098 intcnt 20821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-ntr 20824 |
This theorem is referenced by: iocnct 39767 iccnct 39768 |
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