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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 10689 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | pnfxr 10684 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | 1rp 12381 | . . . 4 ⊢ 1 ∈ ℝ+ | |
4 | pnfinf 30862 | . . . 4 ⊢ (1 ∈ ℝ+ → 1(⋘‘ℝ*𝑠)+∞) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ 1(⋘‘ℝ*𝑠)+∞ |
6 | breq1 5033 | . . . 4 ⊢ (𝑥 = 1 → (𝑥(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)𝑦)) | |
7 | breq2 5034 | . . . 4 ⊢ (𝑦 = +∞ → (1(⋘‘ℝ*𝑠)𝑦 ↔ 1(⋘‘ℝ*𝑠)+∞)) | |
8 | 6, 7 | rspc2ev 3583 | . . 3 ⊢ ((1 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 1(⋘‘ℝ*𝑠)+∞) → ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦) |
9 | 1, 2, 5, 8 | mp3an 1458 | . 2 ⊢ ∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 |
10 | rexnal 3201 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
11 | dfrex2 3202 | . . . 4 ⊢ (∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) | |
12 | 11 | rexbii 3210 | . . 3 ⊢ (∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ∃𝑥 ∈ ℝ* ¬ ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
13 | xrsex 20106 | . . . . 5 ⊢ ℝ*𝑠 ∈ V | |
14 | xrsbas 20107 | . . . . . 6 ⊢ ℝ* = (Base‘ℝ*𝑠) | |
15 | xrs0 30709 | . . . . . 6 ⊢ 0 = (0g‘ℝ*𝑠) | |
16 | eqid 2798 | . . . . . 6 ⊢ (⋘‘ℝ*𝑠) = (⋘‘ℝ*𝑠) | |
17 | 14, 15, 16 | isarchi 30861 | . . . . 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 323 | . . 3 ⊢ (¬ ℝ*𝑠 ∈ Archi ↔ ¬ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* ¬ 𝑥(⋘‘ℝ*𝑠)𝑦) |
20 | 10, 12, 19 | 3bitr4i 306 | . 2 ⊢ (∃𝑥 ∈ ℝ* ∃𝑦 ∈ ℝ* 𝑥(⋘‘ℝ*𝑠)𝑦 ↔ ¬ ℝ*𝑠 ∈ Archi) |
21 | 9, 20 | mpbi 233 | 1 ⊢ ¬ ℝ*𝑠 ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 0cc0 10526 1c1 10527 +∞cpnf 10661 ℝ*cxr 10663 ℝ+crp 12377 ℝ*𝑠cxrs 16765 ⋘cinftm 30855 Archicarchi 30856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12886 df-seq 13365 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-tset 16576 df-ple 16577 df-ds 16579 df-0g 16707 df-xrs 16767 df-plt 17560 df-minusg 18099 df-mulg 18217 df-inftm 30857 df-archi 30858 |
This theorem is referenced by: (None) |
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