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Mathbox for Jim Kingdon |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccioo01 | Structured version Visualization version GIF version |
Description: The closed unit interval is equinumerous to the open unit interval. Based on a Mastodon post by Michael Kinyon. (Contributed by Jim Kingdon, 4-Jun-2024.) |
Ref | Expression |
---|---|
iccioo01 | ⊢ (0[,]1) ≈ (0(,)1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn 12325 | . . . . 5 ⊢ 4 ∈ ℕ | |
2 | nnrecre 12284 | . . . . 5 ⊢ (4 ∈ ℕ → (1 / 4) ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1 / 4) ∈ ℝ |
4 | halfre 12456 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
5 | 2lt4 12417 | . . . . 5 ⊢ 2 < 4 | |
6 | 2re 12316 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 12326 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2pos 12345 | . . . . . 6 ⊢ 0 < 2 | |
9 | 4pos 12349 | . . . . . 6 ⊢ 0 < 4 | |
10 | 6, 7, 8, 9 | ltrecii 12160 | . . . . 5 ⊢ (2 < 4 ↔ (1 / 4) < (1 / 2)) |
11 | 5, 10 | mpbi 229 | . . . 4 ⊢ (1 / 4) < (1 / 2) |
12 | iccen 13506 | . . . 4 ⊢ (((1 / 4) ∈ ℝ ∧ (1 / 2) ∈ ℝ ∧ (1 / 4) < (1 / 2)) → (0[,]1) ≈ ((1 / 4)[,](1 / 2))) | |
13 | 3, 4, 11, 12 | mp3an 1457 | . . 3 ⊢ (0[,]1) ≈ ((1 / 4)[,](1 / 2)) |
14 | ovex 7449 | . . . 4 ⊢ (0(,)1) ∈ V | |
15 | 0xr 11291 | . . . . 5 ⊢ 0 ∈ ℝ* | |
16 | 1xr 11303 | . . . . 5 ⊢ 1 ∈ ℝ* | |
17 | 7, 9 | recgt0ii 12150 | . . . . 5 ⊢ 0 < (1 / 4) |
18 | halflt1 12460 | . . . . 5 ⊢ (1 / 2) < 1 | |
19 | iccssioo 13425 | . . . . 5 ⊢ (((0 ∈ ℝ* ∧ 1 ∈ ℝ*) ∧ (0 < (1 / 4) ∧ (1 / 2) < 1)) → ((1 / 4)[,](1 / 2)) ⊆ (0(,)1)) | |
20 | 15, 16, 17, 18, 19 | mp4an 691 | . . . 4 ⊢ ((1 / 4)[,](1 / 2)) ⊆ (0(,)1) |
21 | ssdomg 9019 | . . . 4 ⊢ ((0(,)1) ∈ V → (((1 / 4)[,](1 / 2)) ⊆ (0(,)1) → ((1 / 4)[,](1 / 2)) ≼ (0(,)1))) | |
22 | 14, 20, 21 | mp2 9 | . . 3 ⊢ ((1 / 4)[,](1 / 2)) ≼ (0(,)1) |
23 | endomtr 9031 | . . 3 ⊢ (((0[,]1) ≈ ((1 / 4)[,](1 / 2)) ∧ ((1 / 4)[,](1 / 2)) ≼ (0(,)1)) → (0[,]1) ≼ (0(,)1)) | |
24 | 13, 22, 23 | mp2an 690 | . 2 ⊢ (0[,]1) ≼ (0(,)1) |
25 | ovex 7449 | . . 3 ⊢ (0[,]1) ∈ V | |
26 | ioossicc 13442 | . . 3 ⊢ (0(,)1) ⊆ (0[,]1) | |
27 | ssdomg 9019 | . . 3 ⊢ ((0[,]1) ∈ V → ((0(,)1) ⊆ (0[,]1) → (0(,)1) ≼ (0[,]1))) | |
28 | 25, 26, 27 | mp2 9 | . 2 ⊢ (0(,)1) ≼ (0[,]1) |
29 | sbth 9116 | . 2 ⊢ (((0[,]1) ≼ (0(,)1) ∧ (0(,)1) ≼ (0[,]1)) → (0[,]1) ≈ (0(,)1)) | |
30 | 24, 28, 29 | mp2an 690 | 1 ⊢ (0[,]1) ≈ (0(,)1) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3463 ⊆ wss 3939 class class class wbr 5143 (class class class)co 7416 ≈ cen 8959 ≼ cdom 8960 ℝcr 11137 0cc0 11138 1c1 11139 ℝ*cxr 11277 < clt 11278 / cdiv 11901 ℕcn 12242 2c2 12297 4c4 12299 (,)cioo 13356 [,]cicc 13359 |
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-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-rp 13007 df-ioo 13360 df-icc 13363 |
This theorem is referenced by: (None) |
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