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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 45769. (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 11301 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 ∈ ℝ*) |
3 | 1xr 11313 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℝ*) |
5 | 0lt1 11776 | . . . . . . . 8 ⊢ 0 < 1 | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 < 1) |
7 | salexct2.3 | . . . . . . 7 ⊢ 𝐵 = (0[,]1) | |
8 | 2, 4, 6, 7 | iccnct 44973 | . . . . . 6 ⊢ (⊤ → ¬ 𝐵 ≼ ω) |
9 | 8 | mptru 1540 | . . . . 5 ⊢ ¬ 𝐵 ≼ ω |
10 | 2re 12326 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
11 | 10 | rexri 11312 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 2 ∈ ℝ*) |
13 | 1lt2 12423 | . . . . . . . . 9 ⊢ 1 < 2 | |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 1 < 2) |
15 | eqid 2728 | . . . . . . . 8 ⊢ (1(,]2) = (1(,]2) | |
16 | 4, 12, 14, 15 | iocnct 44972 | . . . . . . 7 ⊢ (⊤ → ¬ (1(,]2) ≼ ω) |
17 | 16 | mptru 1540 | . . . . . 6 ⊢ ¬ (1(,]2) ≼ ω |
18 | salexct2.1 | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
19 | 18, 7 | difeq12i 4120 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐵) = ((0[,]2) ∖ (0[,]1)) |
20 | 2, 4, 6 | xrltled 13171 | . . . . . . . . . 10 ⊢ (⊤ → 0 ≤ 1) |
21 | 2, 4, 12, 20 | iccdificc 44971 | . . . . . . . . 9 ⊢ (⊤ → ((0[,]2) ∖ (0[,]1)) = (1(,]2)) |
22 | 21 | mptru 1540 | . . . . . . . 8 ⊢ ((0[,]2) ∖ (0[,]1)) = (1(,]2) |
23 | 19, 22 | eqtri 2756 | . . . . . . 7 ⊢ (𝐴 ∖ 𝐵) = (1(,]2) |
24 | 23 | breq1i 5159 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ≼ ω ↔ (1(,]2) ≼ ω) |
25 | 17, 24 | mtbir 322 | . . . . 5 ⊢ ¬ (𝐴 ∖ 𝐵) ≼ ω |
26 | 9, 25 | pm3.2i 469 | . . . 4 ⊢ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω) |
27 | ioran 981 | . . . 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 485 | . 2 ⊢ ¬ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω)) |
30 | breq1 5155 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω)) | |
31 | difeq2 4116 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵)) | |
32 | 31 | breq1d 5162 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝐵) ≼ ω)) |
33 | 30, 32 | orbi12d 916 | . . 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 3687 | . 2 ⊢ (𝐵 ∈ 𝑆 ↔ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω))) |
36 | 29, 35 | mtbir 322 | 1 ⊢ ¬ 𝐵 ∈ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 ∨ wo 845 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 {crab 3430 ∖ cdif 3946 𝒫 cpw 4606 class class class wbr 5152 (class class class)co 7426 ωcom 7878 ≼ cdom 8970 0cc0 11148 1c1 11149 ℝ*cxr 11287 < clt 11288 2c2 12307 (,]cioc 13367 [,]cicc 13369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-omul 8500 df-er 8733 df-map 8855 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-inf 9476 df-oi 9543 df-card 9972 df-acn 9975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-z 12599 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ioc 13371 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-topgen 17434 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-top 22824 df-topon 22841 df-bases 22877 df-ntr 22952 |
This theorem is referenced by: salexct3 45777 salgencntex 45778 salgensscntex 45779 |
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