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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 12243 | . . . . 5 ⊢ 4 ∈ ℕ | |
2 | nnrecre 12202 | . . . . 5 ⊢ (4 ∈ ℕ → (1 / 4) ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1 / 4) ∈ ℝ |
4 | halfre 12374 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
5 | 2lt4 12335 | . . . . 5 ⊢ 2 < 4 | |
6 | 2re 12234 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 12244 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2pos 12263 | . . . . . 6 ⊢ 0 < 2 | |
9 | 4pos 12267 | . . . . . 6 ⊢ 0 < 4 | |
10 | 6, 7, 8, 9 | ltrecii 12078 | . . . . 5 ⊢ (2 < 4 ↔ (1 / 4) < (1 / 2)) |
11 | 5, 10 | mpbi 229 | . . . 4 ⊢ (1 / 4) < (1 / 2) |
12 | iccen 13421 | . . . 4 ⊢ (((1 / 4) ∈ ℝ ∧ (1 / 2) ∈ ℝ ∧ (1 / 4) < (1 / 2)) → (0[,]1) ≈ ((1 / 4)[,](1 / 2))) | |
13 | 3, 4, 11, 12 | mp3an 1462 | . . 3 ⊢ (0[,]1) ≈ ((1 / 4)[,](1 / 2)) |
14 | ovex 7395 | . . . 4 ⊢ (0(,)1) ∈ V | |
15 | 0xr 11209 | . . . . 5 ⊢ 0 ∈ ℝ* | |
16 | 1xr 11221 | . . . . 5 ⊢ 1 ∈ ℝ* | |
17 | 7, 9 | recgt0ii 12068 | . . . . 5 ⊢ 0 < (1 / 4) |
18 | halflt1 12378 | . . . . 5 ⊢ (1 / 2) < 1 | |
19 | iccssioo 13340 | . . . . 5 ⊢ (((0 ∈ ℝ* ∧ 1 ∈ ℝ*) ∧ (0 < (1 / 4) ∧ (1 / 2) < 1)) → ((1 / 4)[,](1 / 2)) ⊆ (0(,)1)) | |
20 | 15, 16, 17, 18, 19 | mp4an 692 | . . . 4 ⊢ ((1 / 4)[,](1 / 2)) ⊆ (0(,)1) |
21 | ssdomg 8947 | . . . 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 8959 | . . 3 ⊢ (((0[,]1) ≈ ((1 / 4)[,](1 / 2)) ∧ ((1 / 4)[,](1 / 2)) ≼ (0(,)1)) → (0[,]1) ≼ (0(,)1)) | |
24 | 13, 22, 23 | mp2an 691 | . 2 ⊢ (0[,]1) ≼ (0(,)1) |
25 | ovex 7395 | . . 3 ⊢ (0[,]1) ∈ V | |
26 | ioossicc 13357 | . . 3 ⊢ (0(,)1) ⊆ (0[,]1) | |
27 | ssdomg 8947 | . . 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 9044 | . 2 ⊢ (((0[,]1) ≼ (0(,)1) ∧ (0(,)1) ≼ (0[,]1)) → (0[,]1) ≈ (0(,)1)) | |
30 | 24, 28, 29 | mp2an 691 | 1 ⊢ (0[,]1) ≈ (0(,)1) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3448 ⊆ wss 3915 class class class wbr 5110 (class class class)co 7362 ≈ cen 8887 ≼ cdom 8888 ℝcr 11057 0cc0 11058 1c1 11059 ℝ*cxr 11195 < clt 11196 / cdiv 11819 ℕcn 12160 2c2 12215 4c4 12217 (,)cioo 13271 [,]cicc 13274 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-rp 12923 df-ioo 13275 df-icc 13278 |
This theorem is referenced by: (None) |
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