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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 11742 | . . . . 5 ⊢ 4 ∈ ℕ | |
2 | nnrecre 11701 | . . . . 5 ⊢ (4 ∈ ℕ → (1 / 4) ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1 / 4) ∈ ℝ |
4 | halfre 11873 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
5 | 2lt4 11834 | . . . . 5 ⊢ 2 < 4 | |
6 | 2re 11733 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 11743 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2pos 11762 | . . . . . 6 ⊢ 0 < 2 | |
9 | 4pos 11766 | . . . . . 6 ⊢ 0 < 4 | |
10 | 6, 7, 8, 9 | ltrecii 11579 | . . . . 5 ⊢ (2 < 4 ↔ (1 / 4) < (1 / 2)) |
11 | 5, 10 | mpbi 233 | . . . 4 ⊢ (1 / 4) < (1 / 2) |
12 | iccen 12914 | . . . 4 ⊢ (((1 / 4) ∈ ℝ ∧ (1 / 2) ∈ ℝ ∧ (1 / 4) < (1 / 2)) → (0[,]1) ≈ ((1 / 4)[,](1 / 2))) | |
13 | 3, 4, 11, 12 | mp3an 1459 | . . 3 ⊢ (0[,]1) ≈ ((1 / 4)[,](1 / 2)) |
14 | ovex 7176 | . . . 4 ⊢ (0(,)1) ∈ V | |
15 | 0xr 10711 | . . . . 5 ⊢ 0 ∈ ℝ* | |
16 | 1xr 10723 | . . . . 5 ⊢ 1 ∈ ℝ* | |
17 | 7, 9 | recgt0ii 11569 | . . . . 5 ⊢ 0 < (1 / 4) |
18 | halflt1 11877 | . . . . 5 ⊢ (1 / 2) < 1 | |
19 | iccssioo 12833 | . . . . 5 ⊢ (((0 ∈ ℝ* ∧ 1 ∈ ℝ*) ∧ (0 < (1 / 4) ∧ (1 / 2) < 1)) → ((1 / 4)[,](1 / 2)) ⊆ (0(,)1)) | |
20 | 15, 16, 17, 18, 19 | mp4an 693 | . . . 4 ⊢ ((1 / 4)[,](1 / 2)) ⊆ (0(,)1) |
21 | ssdomg 8566 | . . . 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 8578 | . . 3 ⊢ (((0[,]1) ≈ ((1 / 4)[,](1 / 2)) ∧ ((1 / 4)[,](1 / 2)) ≼ (0(,)1)) → (0[,]1) ≼ (0(,)1)) | |
24 | 13, 22, 23 | mp2an 692 | . 2 ⊢ (0[,]1) ≼ (0(,)1) |
25 | ovex 7176 | . . 3 ⊢ (0[,]1) ∈ V | |
26 | ioossicc 12850 | . . 3 ⊢ (0(,)1) ⊆ (0[,]1) | |
27 | ssdomg 8566 | . . 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 8651 | . 2 ⊢ (((0[,]1) ≼ (0(,)1) ∧ (0(,)1) ≼ (0[,]1)) → (0[,]1) ≈ (0(,)1)) | |
30 | 24, 28, 29 | mp2an 692 | 1 ⊢ (0[,]1) ≈ (0(,)1) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 Vcvv 3407 ⊆ wss 3854 class class class wbr 5025 (class class class)co 7143 ≈ cen 8517 ≼ cdom 8518 ℝcr 10559 0cc0 10560 1c1 10561 ℝ*cxr 10697 < clt 10698 / cdiv 11320 ℕcn 11659 2c2 11714 4c4 11716 (,)cioo 12764 [,]cicc 12767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rmo 3076 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-div 11321 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-rp 12416 df-ioo 12768 df-icc 12771 |
This theorem is referenced by: (None) |
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