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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 45464 | . . . 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 13445 | . . . . 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 5155 | . . . . . . 7 ⊢ (𝐶 ≼ ω ↔ (𝐴(,)𝐵) ≼ ω) |
7 | 6 | biimpi 216 | . . . . . 6 ⊢ (𝐶 ≼ ω → (𝐴(,)𝐵) ≼ ω) |
8 | nnenom 14018 | . . . . . . . 8 ⊢ ℕ ≈ ω | |
9 | 8 | ensymi 9043 | . . . . . . 7 ⊢ ω ≈ ℕ |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝐶 ≼ ω → ω ≈ ℕ) |
11 | domentr 9052 | . . . . . 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 24868 | . . . 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 2777 | . 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 13410 | . . . . . 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 2943 | . . 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 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≈ cen 8981 ≼ cdom 8982 ℝcr 11152 ℝ*cxr 11292 < clt 11293 ℕcn 12264 (,)cioo 13384 topGenctg 17484 intcnt 23041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-ntr 23044 |
This theorem is referenced by: iocnct 45493 iccnct 45494 |
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