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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrnarchi | Structured version Visualization version GIF version |
Description: The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.) |
Ref | Expression |
---|---|
xrnarchi | ⊢ ¬ ℝ*𝑠 ∈ Archi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1xr 11224 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 11219 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | 1rp 12929 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | pnfinf 32090 | . . . 4 ⊢ (1 ∈ ℝ+ → 1(⋘‘ℝ*𝑠)+∞) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1(⋘‘ℝ*𝑠)+∞ |
6 | breq1 5114 | . . . 4 ⊢ (𝑥 = 1 → (𝑥(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)𝑦)) | |
7 | breq2 5115 | . . . 4 ⊢ (𝑦 = +∞ → (1(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)+∞)) | |
8 | 6, 7 | rspc2ev 3594 | . . 3 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 1(⋘‘ℝ*𝑠)+∞) → ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦) |
9 | 1, 2, 5, 8 | mp3an 1462 | . 2 ⊢ ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 |
10 | rexnal 3100 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
11 | dfrex2 3073 | . . . 4 ⊢ (∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
12 | 11 | rexbii 3094 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
13 | xrsex 20850 | . . . . 5 ⊢ ℝ*𝑠 ∈ V | |
14 | xrsbas 20851 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | xrs0 31937 | . . . . . 6 ⊢ 0 = (0g‘ℝ*𝑠) | |
16 | eqid 2732 | . . . . . 6 ⊢ (⋘‘ℝ*𝑠) = (⋘‘ℝ*𝑠) | |
17 | 14, 15, 16 | isarchi 32089 | . . . . 5 ⊢ (ℝ*𝑠 ∈ V → (ℝ*𝑠 ∈ Archi ↔ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦)) |
18 | 13, 17 | ax-mp 5 | . . . 4 ⊢ (ℝ*𝑠 ∈ Archi ↔ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
19 | 18 | notbii 320 | . . 3 ⊢ (¬ ℝ*𝑠 ∈ Archi ↔ ¬ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
20 | 10, 12, 19 | 3bitr4i 303 | . 2 ⊢ (∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ℝ*𝑠 ∈ Archi) |
21 | 9, 20 | mpbi 229 | 1 ⊢ ¬ ℝ*𝑠 ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∈ wcel 2107 ∀wral 3061 ∃wrex 3070 Vcvv 3447 class class class wbr 5111 ‘cfv 6502 0cc0 11061 1c1 11062 +∞cpnf 11196 ℝ*cxr 11198 ℝ+crp 12925 ℝ*𝑠cxrs 17397 ⋘cinftm 32083 Archicarchi 32084 |
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 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-mulcom 11125 ax-addass 11126 ax-mulass 11127 ax-distr 11128 ax-i2m1 11129 ax-1ne0 11130 ax-1rid 11131 ax-rnegex 11132 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 ax-pre-ltadd 11137 ax-pre-mulgt0 11138 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-tr 5229 df-id 5537 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5594 df-we 5596 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-pred 6259 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7319 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7809 df-1st 7927 df-2nd 7928 df-frecs 8218 df-wrecs 8249 df-recs 8323 df-rdg 8362 df-1o 8418 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-fin 8895 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-sub 11397 df-neg 11398 df-nn 12164 df-2 12226 df-3 12227 df-4 12228 df-5 12229 df-6 12230 df-7 12231 df-8 12232 df-9 12233 df-n0 12424 df-z 12510 df-dec 12629 df-uz 12774 df-rp 12926 df-xneg 13043 df-xadd 13044 df-xmul 13045 df-fz 13436 df-seq 13918 df-struct 17031 df-slot 17066 df-ndx 17078 df-base 17096 df-plusg 17161 df-mulr 17162 df-tset 17167 df-ple 17168 df-ds 17170 df-0g 17338 df-xrs 17399 df-plt 18234 df-minusg 18767 df-mulg 18888 df-inftm 32085 df-archi 32086 |
This theorem is referenced by: (None) |
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