| 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 12349 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 2 | nnrecre 12308 | . . . . 5 ⊢ (4 ∈ ℕ → (1 / 4) ∈ ℝ) | |
| 3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (1 / 4) ∈ ℝ |
| 4 | halfre 12480 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
| 5 | 2lt4 12441 | . . . . 5 ⊢ 2 < 4 | |
| 6 | 2re 12340 | . . . . . 6 ⊢ 2 ∈ ℝ | |
| 7 | 4re 12350 | . . . . . 6 ⊢ 4 ∈ ℝ | |
| 8 | 2pos 12369 | . . . . . 6 ⊢ 0 < 2 | |
| 9 | 4pos 12373 | . . . . . 6 ⊢ 0 < 4 | |
| 10 | 6, 7, 8, 9 | ltrecii 12184 | . . . . 5 ⊢ (2 < 4 ↔ (1 / 4) < (1 / 2)) |
| 11 | 5, 10 | mpbi 230 | . . . 4 ⊢ (1 / 4) < (1 / 2) |
| 12 | iccen 13537 | . . . 4 ⊢ (((1 / 4) ∈ ℝ ∧ (1 / 2) ∈ ℝ ∧ (1 / 4) < (1 / 2)) → (0[,]1) ≈ ((1 / 4)[,](1 / 2))) | |
| 13 | 3, 4, 11, 12 | mp3an 1463 | . . 3 ⊢ (0[,]1) ≈ ((1 / 4)[,](1 / 2)) |
| 14 | ovex 7464 | . . . 4 ⊢ (0(,)1) ∈ V | |
| 15 | 0xr 11308 | . . . . 5 ⊢ 0 ∈ ℝ* | |
| 16 | 1xr 11320 | . . . . 5 ⊢ 1 ∈ ℝ* | |
| 17 | 7, 9 | recgt0ii 12174 | . . . . 5 ⊢ 0 < (1 / 4) |
| 18 | halflt1 12484 | . . . . 5 ⊢ (1 / 2) < 1 | |
| 19 | iccssioo 13456 | . . . . 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 9040 | . . . 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 9052 | . . 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 7464 | . . 3 ⊢ (0[,]1) ∈ V | |
| 26 | ioossicc 13473 | . . 3 ⊢ (0(,)1) ⊆ (0[,]1) | |
| 27 | ssdomg 9040 | . . 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 9133 | . 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 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 ≈ cen 8982 ≼ cdom 8983 ℝcr 11154 0cc0 11155 1c1 11156 ℝ*cxr 11294 < clt 11295 / cdiv 11920 ℕcn 12266 2c2 12321 4c4 12323 (,)cioo 13387 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-rp 13035 df-ioo 13391 df-icc 13394 |
| This theorem is referenced by: (None) |
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