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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 10694 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 10689 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | 1rp 12387 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | pnfinf 30807 | . . . 4 ⊢ (1 ∈ ℝ+ → 1(⋘‘ℝ*𝑠)+∞) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1(⋘‘ℝ*𝑠)+∞ |
6 | breq1 5061 | . . . 4 ⊢ (𝑥 = 1 → (𝑥(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)𝑦)) | |
7 | breq2 5062 | . . . 4 ⊢ (𝑦 = +∞ → (1(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)+∞)) | |
8 | 6, 7 | rspc2ev 3634 | . . 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 20554 | . . . . 5 ⊢ ℝ*𝑠 ∈ V | |
14 | xrsbas 20555 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | xrs0 30657 | . . . . . 6 ⊢ 0 = (0g‘ℝ*𝑠) | |
16 | eqid 2821 | . . . . . 6 ⊢ (⋘‘ℝ*𝑠) = (⋘‘ℝ*𝑠) | |
17 | 14, 15, 16 | isarchi 30806 | . . . . 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 2110 ∀wral 3138 ∃wrex 3139 Vcvv 3494 class class class wbr 5058 ‘cfv 6349 0cc0 10531 1c1 10532 +∞cpnf 10666 ℝ*cxr 10668 ℝ+crp 12383 ℝ*𝑠cxrs 16767 ⋘cinftm 30800 Archicarchi 30801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-fz 12887 df-seq 13364 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-tset 16578 df-ple 16579 df-ds 16581 df-0g 16709 df-xrs 16769 df-plt 17562 df-minusg 18101 df-mulg 18219 df-inftm 30802 df-archi 30803 |
This theorem is referenced by: (None) |
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