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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iocnct | Structured version Visualization version GIF version |
Description: A non empty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
iocnct.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
iocnct.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
iocnct.l | ⊢ (𝜑 → 𝐴 < 𝐵) |
iocnct.c | ⊢ 𝐶 = (𝐴(,]𝐵) |
Ref | Expression |
---|---|
iocnct | ⊢ (𝜑 → ¬ 𝐶 ≼ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iocnct.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | iocnct.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
3 | iocnct.l | . . 3 ⊢ (𝜑 → 𝐴 < 𝐵) | |
4 | eqid 2770 | . . 3 ⊢ (𝐴(,)𝐵) = (𝐴(,)𝐵) | |
5 | 1, 2, 3, 4 | ioonct 40276 | . 2 ⊢ (𝜑 → ¬ (𝐴(,)𝐵) ≼ ω) |
6 | ioossioc 40228 | . . . 4 ⊢ (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵) | |
7 | iocnct.c | . . . 4 ⊢ 𝐶 = (𝐴(,]𝐵) | |
8 | 6, 7 | sseqtr4i 3785 | . . 3 ⊢ (𝐴(,)𝐵) ⊆ 𝐶 |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐶) |
10 | 5, 9 | ssnct 39764 | 1 ⊢ (𝜑 → ¬ 𝐶 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1630 ∈ wcel 2144 ⊆ wss 3721 class class class wbr 4784 (class class class)co 6792 ωcom 7211 ≼ cdom 8106 ℝ*cxr 10274 < clt 10275 (,)cioo 12379 (,]cioc 12380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-2o 7713 df-oadd 7716 df-omul 7717 df-er 7895 df-map 8010 df-pm 8011 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-sup 8503 df-inf 8504 df-oi 8570 df-card 8964 df-acn 8967 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-n0 11494 df-z 11579 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-ioc 12384 df-ico 12385 df-icc 12386 df-fz 12533 df-fzo 12673 df-fl 12800 df-seq 13008 df-exp 13067 df-hash 13321 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-limsup 14409 df-clim 14426 df-rlim 14427 df-sum 14624 df-topgen 16311 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-top 20918 df-topon 20935 df-bases 20970 df-ntr 21044 |
This theorem is referenced by: salexct2 41068 |
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