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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 11280 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 11275 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | 1rp 12985 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | pnfinf 32765 | . . . 4 ⊢ (1 ∈ ℝ+ → 1(⋘‘ℝ*𝑠)+∞) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1(⋘‘ℝ*𝑠)+∞ |
6 | breq1 5151 | . . . 4 ⊢ (𝑥 = 1 → (𝑥(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)𝑦)) | |
7 | breq2 5152 | . . . 4 ⊢ (𝑦 = +∞ → (1(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)+∞)) | |
8 | 6, 7 | rspc2ev 3624 | . . 3 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 1(⋘‘ℝ*𝑠)+∞) → ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦) |
9 | 1, 2, 5, 8 | mp3an 1460 | . 2 ⊢ ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 |
10 | rexnal 3099 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
11 | dfrex2 3072 | . . . 4 ⊢ (∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
12 | 11 | rexbii 3093 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
13 | xrsex 21249 | . . . . 5 ⊢ ℝ*𝑠 ∈ V | |
14 | xrsbas 21250 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | xrs0 32609 | . . . . . 6 ⊢ 0 = (0g‘ℝ*𝑠) | |
16 | eqid 2731 | . . . . . 6 ⊢ (⋘‘ℝ*𝑠) = (⋘‘ℝ*𝑠) | |
17 | 14, 15, 16 | isarchi 32764 | . . . . 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 2105 ∀wral 3060 ∃wrex 3069 Vcvv 3473 class class class wbr 5148 ‘cfv 6543 0cc0 11116 1c1 11117 +∞cpnf 11252 ℝ*cxr 11254 ℝ+crp 12981 ℝ*𝑠cxrs 17453 ⋘cinftm 32758 Archicarchi 32759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-fz 13492 df-seq 13974 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-tset 17223 df-ple 17224 df-ds 17226 df-0g 17394 df-xrs 17455 df-plt 18293 df-minusg 18865 df-mulg 18994 df-inftm 32760 df-archi 32761 |
This theorem is referenced by: (None) |
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