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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salgensscntex | Structured version Visualization version GIF version |
Description: This counterexample shows that the sigma-algebra generated by a set is not the smallest sigma-algebra containing the set, if we consider also sigma-algebras with a larger base set. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
salgensscntex.a | ⊢ 𝐴 = (0[,]2) |
salgensscntex.s | ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} |
salgensscntex.x | ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
salgensscntex.g | ⊢ 𝐺 = (SalGen‘𝑋) |
Ref | Expression |
---|---|
salgensscntex | ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salgensscntex.x | . . 3 ⊢ 𝑋 = ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
2 | 0re 9919 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
3 | 2re 10967 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
4 | 2, 3 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
5 | 2 | leidi 10441 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
6 | 1le2 11118 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
7 | 5, 6 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
8 | iccss 12112 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
9 | 4, 7, 8 | mp2an 704 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
10 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
11 | 9, 10 | sseldi 3566 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
12 | salgensscntex.a | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
13 | 11, 12 | syl6eleqr 2699 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
14 | snelpwi 4839 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
16 | snfi 7923 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
17 | fict 8433 | . . . . . . . . . 10 ⊢ ({𝑦} ∈ Fin → {𝑦} ≼ ω) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ {𝑦} ≼ ω |
19 | orc 399 | . . . . . . . . 9 ⊢ ({𝑦} ≼ ω → ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω)) | |
20 | 18, 19 | ax-mp 5 | . . . . . . . 8 ⊢ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω) |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω)) |
22 | 15, 21 | jca 553 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
23 | breq1 4586 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
24 | difeq2 3684 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
25 | 24 | breq1d 4593 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ {𝑦}) ≼ ω)) |
26 | 23, 25 | orbi12d 742 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → ((𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω) ↔ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
27 | salgensscntex.s | . . . . . . 7 ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} | |
28 | 26, 27 | elrab2 3333 | . . . . . 6 ⊢ ({𝑦} ∈ 𝑆 ↔ ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
29 | 22, 28 | sylibr 223 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝑆) |
30 | 29 | rgen 2906 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
31 | eqid 2610 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
32 | 31 | rnmptss 6299 | . . . 4 ⊢ (∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 → ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ⊆ 𝑆) |
33 | 30, 32 | ax-mp 5 | . . 3 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ⊆ 𝑆 |
34 | 1, 33 | eqsstri 3598 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
35 | ovex 6577 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
36 | 12, 35 | eqeltri 2684 | . . . . 5 ⊢ 𝐴 ∈ V |
37 | 36 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ∈ V) |
38 | 37, 27 | salexct 39228 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
39 | 38 | trud 1484 | . 2 ⊢ 𝑆 ∈ SAlg |
40 | ovex 6577 | . . . . . . . . 9 ⊢ (0[,]1) ∈ V | |
41 | 40 | mptex 6390 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
42 | 41 | rnex 6992 | . . . . . . 7 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
43 | 1, 42 | eqeltri 2684 | . . . . . 6 ⊢ 𝑋 ∈ V |
44 | 43 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝑋 ∈ V) |
45 | salgensscntex.g | . . . . 5 ⊢ 𝐺 = (SalGen‘𝑋) | |
46 | 1 | unieqi 4381 | . . . . . 6 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
47 | snex 4835 | . . . . . . . . 9 ⊢ {𝑦} ∈ V | |
48 | 47 | rgenw 2908 | . . . . . . . 8 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
49 | dfiun3g 5299 | . . . . . . . 8 ⊢ (∀𝑦 ∈ (0[,]1){𝑦} ∈ V → ∪ 𝑦 ∈ (0[,]1){𝑦} = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦})) | |
50 | 48, 49 | ax-mp 5 | . . . . . . 7 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
51 | 50 | eqcomi 2619 | . . . . . 6 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
52 | iunid 4511 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
53 | 46, 51, 52 | 3eqtrri 2637 | . . . . 5 ⊢ (0[,]1) = ∪ 𝑋 |
54 | 44, 45, 53 | unisalgen 39234 | . . . 4 ⊢ (⊤ → (0[,]1) ∈ 𝐺) |
55 | 54 | trud 1484 | . . 3 ⊢ (0[,]1) ∈ 𝐺 |
56 | eqid 2610 | . . . 4 ⊢ (0[,]1) = (0[,]1) | |
57 | 12, 27, 56 | salexct2 39233 | . . 3 ⊢ ¬ (0[,]1) ∈ 𝑆 |
58 | nelss 3627 | . . 3 ⊢ (((0[,]1) ∈ 𝐺 ∧ ¬ (0[,]1) ∈ 𝑆) → ¬ 𝐺 ⊆ 𝑆) | |
59 | 55, 57, 58 | mp2an 704 | . 2 ⊢ ¬ 𝐺 ⊆ 𝑆 |
60 | 34, 39, 59 | 3pm3.2i 1232 | 1 ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≼ cdom 7839 Fincfn 7841 ℝcr 9814 0cc0 9815 1c1 9816 ≤ cle 9954 2c2 10947 [,]cicc 12049 SAlgcsalg 39204 SalGencsalgen 39208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-ntr 20634 df-salg 39205 df-salgen 39209 |
This theorem is referenced by: (None) |
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