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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 45034 | . . . 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 13420 | . . . . 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 5156 | . . . . . . 7 ⊢ (𝐶 ≼ ω ↔ (𝐴(,)𝐵) ≼ ω) |
7 | 6 | biimpi 215 | . . . . . 6 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ω) |
8 | nnenom 13981 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
9 | 8 | ensymi 9025 | . . . . . . 7 ⊢ ω ≈ ℕ |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝐶 ≼ ω → ω ≈ ℕ) |
11 | domentr 9034 | . . . . . 6 ⊢ (((𝐴(,)𝐵) ≼ ω ∧ ω ≈ ℕ) → (𝐴(,)𝐵) ≼ ℕ) | |
12 | 7, 10, 11 | syl2anc 582 | . . . . 5 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ℕ) |
13 | 12 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → (𝐴(,)𝐵) ≼ ℕ) |
14 | rectbntr0 24792 | . . . 4 ⊢ (((𝐴(,)𝐵) ⊆ ℝ ∧ (𝐴(,)𝐵) ≼ ℕ) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) | |
15 | 4, 13, 14 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) |
16 | 2, 15 | eqtr3d 2767 | . 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 13385 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) | |
21 | 18, 19, 20 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) |
22 | 17, 21 | mpbird 256 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅) |
23 | 22 | neneqd 2934 | . . 3 ⊢ (𝜑 → ¬ (𝐴(,)𝐵) = ∅) |
24 | 23 | adantr 479 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ¬ (𝐴(,)𝐵) = ∅) |
25 | 16, 24 | pm2.65da 815 | 1 ⊢ (𝜑 → ¬ 𝐶 ≼ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ⊆ wss 3944 ∅c0 4322 class class class wbr 5149 ran crn 5679 ‘cfv 6549 (class class class)co 7419 ωcom 7871 ≈ cen 8961 ≼ cdom 8962 ℝcr 11139 ℝ*cxr 11279 < clt 11280 ℕcn 12245 (,)cioo 13359 topGenctg 17422 intcnt 22965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-acn 9967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22840 df-topon 22857 df-bases 22893 df-ntr 22968 |
This theorem is referenced by: iocnct 45063 iccnct 45064 |
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