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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salexct2 | Structured version Visualization version GIF version |
Description: An example of a subset that does not belong to a nontrivial sigma-algebra, see salexct 45713. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
salexct2.1 | ⊢ 𝐴 = (0[,]2) |
salexct2.2 | ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} |
salexct2.3 | ⊢ 𝐵 = (0[,]1) |
Ref | Expression |
---|---|
salexct2 | ⊢ ¬ 𝐵 ∈ 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11286 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 ∈ ℝ*) |
3 | 1xr 11298 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℝ*) |
5 | 0lt1 11761 | . . . . . . . 8 ⊢ 0 < 1 | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 < 1) |
7 | salexct2.3 | . . . . . . 7 ⊢ 𝐵 = (0[,]1) | |
8 | 2, 4, 6, 7 | iccnct 44917 | . . . . . 6 ⊢ (⊤ → ¬ 𝐵 ≼ ω) |
9 | 8 | mptru 1541 | . . . . 5 ⊢ ¬ 𝐵 ≼ ω |
10 | 2re 12311 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
11 | 10 | rexri 11297 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 2 ∈ ℝ*) |
13 | 1lt2 12408 | . . . . . . . . 9 ⊢ 1 < 2 | |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 1 < 2) |
15 | eqid 2728 | . . . . . . . 8 ⊢ (1(,]2) = (1(,]2) | |
16 | 4, 12, 14, 15 | iocnct 44916 | . . . . . . 7 ⊢ (⊤ → ¬ (1(,]2) ≼ ω) |
17 | 16 | mptru 1541 | . . . . . 6 ⊢ ¬ (1(,]2) ≼ ω |
18 | salexct2.1 | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
19 | 18, 7 | difeq12i 4117 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐵) = ((0[,]2) ∖ (0[,]1)) |
20 | 2, 4, 6 | xrltled 13156 | . . . . . . . . . 10 ⊢ (⊤ → 0 ≤ 1) |
21 | 2, 4, 12, 20 | iccdificc 44915 | . . . . . . . . 9 ⊢ (⊤ → ((0[,]2) ∖ (0[,]1)) = (1(,]2)) |
22 | 21 | mptru 1541 | . . . . . . . 8 ⊢ ((0[,]2) ∖ (0[,]1)) = (1(,]2) |
23 | 19, 22 | eqtri 2756 | . . . . . . 7 ⊢ (𝐴 ∖ 𝐵) = (1(,]2) |
24 | 23 | breq1i 5150 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ≼ ω ↔ (1(,]2) ≼ ω) |
25 | 17, 24 | mtbir 323 | . . . . 5 ⊢ ¬ (𝐴 ∖ 𝐵) ≼ ω |
26 | 9, 25 | pm3.2i 470 | . . . 4 ⊢ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω) |
27 | ioran 982 | . . . 4 ⊢ (¬ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω) ↔ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω)) | |
28 | 26, 27 | mpbir 230 | . . 3 ⊢ ¬ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω) |
29 | 28 | intnan 486 | . 2 ⊢ ¬ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω)) |
30 | breq1 5146 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω)) | |
31 | difeq2 4113 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵)) | |
32 | 31 | breq1d 5153 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝐵) ≼ ω)) |
33 | 30, 32 | orbi12d 917 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω) ↔ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω))) |
34 | salexct2.2 | . . 3 ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} | |
35 | 33, 34 | elrab2 3684 | . 2 ⊢ (𝐵 ∈ 𝑆 ↔ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω))) |
36 | 29, 35 | mtbir 323 | 1 ⊢ ¬ 𝐵 ∈ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 846 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 {crab 3428 ∖ cdif 3942 𝒫 cpw 4599 class class class wbr 5143 (class class class)co 7415 ωcom 7865 ≼ cdom 8956 0cc0 11133 1c1 11134 ℝ*cxr 11272 < clt 11273 2c2 12292 (,]cioc 13352 [,]cicc 13354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-oadd 8485 df-omul 8486 df-er 8719 df-map 8841 df-pm 8842 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-acn 9960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660 df-topgen 17419 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-top 22790 df-topon 22807 df-bases 22843 df-ntr 22918 |
This theorem is referenced by: salexct3 45721 salgencntex 45722 salgensscntex 45723 |
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