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 10686 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 10681 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | 1rp 12380 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | pnfinf 30819 | . . . 4 ⊢ (1 ∈ ℝ+ → 1(⋘‘ℝ*𝑠)+∞) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1(⋘‘ℝ*𝑠)+∞ |
6 | breq1 5055 | . . . 4 ⊢ (𝑥 = 1 → (𝑥(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)𝑦)) | |
7 | breq2 5056 | . . . 4 ⊢ (𝑦 = +∞ → (1(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)+∞)) | |
8 | 6, 7 | rspc2ev 3627 | . . 3 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 1(⋘‘ℝ*𝑠)+∞) → ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦) |
9 | 1, 2, 5, 8 | mp3an 1457 | . 2 ⊢ ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 |
10 | rexnal 3238 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
11 | dfrex2 3239 | . . . 4 ⊢ (∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
12 | 11 | rexbii 3247 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
13 | xrsex 20543 | . . . . 5 ⊢ ℝ*𝑠 ∈ V | |
14 | xrsbas 20544 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | xrs0 30669 | . . . . . 6 ⊢ 0 = (0g‘ℝ*𝑠) | |
16 | eqid 2821 | . . . . . 6 ⊢ (⋘‘ℝ*𝑠) = (⋘‘ℝ*𝑠) | |
17 | 14, 15, 16 | isarchi 30818 | . . . . 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 322 | . . 3 ⊢ (¬ ℝ*𝑠 ∈ Archi ↔ ¬ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
20 | 10, 12, 19 | 3bitr4i 305 | . 2 ⊢ (∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ℝ*𝑠 ∈ Archi) |
21 | 9, 20 | mpbi 232 | 1 ⊢ ¬ ℝ*𝑠 ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3486 class class class wbr 5052 ‘cfv 6341 0cc0 10523 1c1 10524 +∞cpnf 10658 ℝ*cxr 10660 ℝ+crp 12376 ℝ*𝑠cxrs 16756 ⋘cinftm 30812 Archicarchi 30813 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-rp 12377 df-xneg 12494 df-xadd 12495 df-xmul 12496 df-fz 12883 df-seq 13360 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-plusg 16561 df-mulr 16562 df-tset 16567 df-ple 16568 df-ds 16570 df-0g 16698 df-xrs 16758 df-plt 17551 df-minusg 18090 df-mulg 18208 df-inftm 30814 df-archi 30815 |
This theorem is referenced by: (None) |
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