![]() |
Mathbox for Jim Kingdon |
< Previous
Next >
Nearby theorems |
|
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 12299 | . . . . 5 ⊢ 4 ∈ ℕ | |
2 | nnrecre 12258 | . . . . 5 ⊢ (4 ∈ ℕ → (1 / 4) ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1 / 4) ∈ ℝ |
4 | halfre 12430 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
5 | 2lt4 12391 | . . . . 5 ⊢ 2 < 4 | |
6 | 2re 12290 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 4re 12300 | . . . . . 6 ⊢ 4 ∈ ℝ | |
8 | 2pos 12319 | . . . . . 6 ⊢ 0 < 2 | |
9 | 4pos 12323 | . . . . . 6 ⊢ 0 < 4 | |
10 | 6, 7, 8, 9 | ltrecii 12134 | . . . . 5 ⊢ (2 < 4 ↔ (1 / 4) < (1 / 2)) |
11 | 5, 10 | mpbi 229 | . . . 4 ⊢ (1 / 4) < (1 / 2) |
12 | iccen 13480 | . . . 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 7438 | . . . 4 ⊢ (0(,)1) ∈ V | |
15 | 0xr 11265 | . . . . 5 ⊢ 0 ∈ ℝ* | |
16 | 1xr 11277 | . . . . 5 ⊢ 1 ∈ ℝ* | |
17 | 7, 9 | recgt0ii 12124 | . . . . 5 ⊢ 0 < (1 / 4) |
18 | halflt1 12434 | . . . . 5 ⊢ (1 / 2) < 1 | |
19 | iccssioo 13399 | . . . . 5 ⊢ (((0 ∈ ℝ* ∧ 1 ∈ ℝ*) ∧ (0 < (1 / 4) ∧ (1 / 2) < 1)) → ((1 / 4)[,](1 / 2)) ⊆ (0(,)1)) | |
20 | 15, 16, 17, 18, 19 | mp4an 690 | . . . 4 ⊢ ((1 / 4)[,](1 / 2)) ⊆ (0(,)1) |
21 | ssdomg 8998 | . . . 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 9010 | . . 3 ⊢ (((0[,]1) ≈ ((1 / 4)[,](1 / 2)) ∧ ((1 / 4)[,](1 / 2)) ≼ (0(,)1)) → (0[,]1) ≼ (0(,)1)) | |
24 | 13, 22, 23 | mp2an 689 | . 2 ⊢ (0[,]1) ≼ (0(,)1) |
25 | ovex 7438 | . . 3 ⊢ (0[,]1) ∈ V | |
26 | ioossicc 13416 | . . 3 ⊢ (0(,)1) ⊆ (0[,]1) | |
27 | ssdomg 8998 | . . 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 9095 | . 2 ⊢ (((0[,]1) ≼ (0(,)1) ∧ (0(,)1) ≼ (0[,]1)) → (0[,]1) ≈ (0(,)1)) | |
30 | 24, 28, 29 | mp2an 689 | 1 ⊢ (0[,]1) ≈ (0(,)1) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 class class class wbr 5141 (class class class)co 7405 ≈ cen 8938 ≼ cdom 8939 ℝcr 11111 0cc0 11112 1c1 11113 ℝ*cxr 11251 < clt 11252 / cdiv 11875 ℕcn 12216 2c2 12271 4c4 12273 (,)cioo 13330 [,]cicc 13333 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-rp 12981 df-ioo 13334 df-icc 13337 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |