| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| 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 45524 | . . . 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 13448 | . . . . 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 5150 | . . . . . . 7 ⊢ (𝐶 ≼ ω ↔ (𝐴(,)𝐵) ≼ ω) |
| 7 | 6 | biimpi 216 | . . . . . 6 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ω) |
| 8 | nnenom 14021 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
| 9 | 8 | ensymi 9044 | . . . . . . 7 ⊢ ω ≈ ℕ |
| 10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝐶 ≼ ω → ω ≈ ℕ) |
| 11 | domentr 9053 | . . . . . 6 ⊢ (((𝐴(,)𝐵) ≼ ω ∧ ω ≈ ℕ) → (𝐴(,)𝐵) ≼ ℕ) | |
| 12 | 7, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ℕ) |
| 13 | 12 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → (𝐴(,)𝐵) ≼ ℕ) |
| 14 | rectbntr0 24854 | . . . 4 ⊢ (((𝐴(,)𝐵) ⊆ ℝ ∧ (𝐴(,)𝐵) ≼ ℕ) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) | |
| 15 | 4, 13, 14 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ((int‘(topGen‘ran (,)))‘(𝐴(,)𝐵)) = ∅) |
| 16 | 2, 15 | eqtr3d 2779 | . 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 13413 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) | |
| 21 | 18, 19, 20 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((𝐴(,)𝐵) ≠ ∅ ↔ 𝐴 < 𝐵)) |
| 22 | 17, 21 | mpbird 257 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅) |
| 23 | 22 | neneqd 2945 | . . 3 ⊢ (𝜑 → ¬ (𝐴(,)𝐵) = ∅) |
| 24 | 23 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝐶 ≼ ω) → ¬ (𝐴(,)𝐵) = ∅) |
| 25 | 16, 24 | pm2.65da 817 | 1 ⊢ (𝜑 → ¬ 𝐶 ≼ ω) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 ℝcr 11154 ℝ*cxr 11294 < clt 11295 ℕcn 12266 (,)cioo 13387 topGenctg 17482 intcnt 23025 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-ntr 23028 |
| This theorem is referenced by: iocnct 45553 iccnct 45554 |
| Copyright terms: Public domain | W3C validator |