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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 11261 | . . . . . . . . . . . 12 ⊢ 0 ∈ ℝ | |
3 | 2re 12338 | . . . . . . . . . . . 12 ⊢ 2 ∈ ℝ | |
4 | 2, 3 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ∈ ℝ ∧ 2 ∈ ℝ) |
5 | 2 | leidi 11795 | . . . . . . . . . . . 12 ⊢ 0 ≤ 0 |
6 | 1le2 12473 | . . . . . . . . . . . 12 ⊢ 1 ≤ 2 | |
7 | 5, 6 | pm3.2i 470 | . . . . . . . . . . 11 ⊢ (0 ≤ 0 ∧ 1 ≤ 2) |
8 | iccss 13452 | . . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 2 ∈ ℝ) ∧ (0 ≤ 0 ∧ 1 ≤ 2)) → (0[,]1) ⊆ (0[,]2)) | |
9 | 4, 7, 8 | mp2an 692 | . . . . . . . . . 10 ⊢ (0[,]1) ⊆ (0[,]2) |
10 | id 22 | . . . . . . . . . 10 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]1)) | |
11 | 9, 10 | sselid 3993 | . . . . . . . . 9 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ (0[,]2)) |
12 | salgensscntex.a | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
13 | 11, 12 | eleqtrrdi 2850 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) → 𝑦 ∈ 𝐴) |
14 | snelpwi 5454 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ∈ 𝒫 𝐴) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝒫 𝐴) |
16 | snfi 9082 | . . . . . . . . . 10 ⊢ {𝑦} ∈ Fin | |
17 | fict 9691 | . . . . . . . . . 10 ⊢ ({𝑦} ∈ Fin → {𝑦} ≼ ω) | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . 9 ⊢ {𝑦} ≼ ω |
19 | orc 867 | . . . . . . . . 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 511 | . . . . . 6 ⊢ (𝑦 ∈ (0[,]1) → ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
23 | breq1 5151 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ≼ ω ↔ {𝑦} ≼ ω)) | |
24 | difeq2 4130 | . . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝐴 ∖ 𝑥) = (𝐴 ∖ {𝑦})) | |
25 | 24 | breq1d 5158 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ {𝑦}) ≼ ω)) |
26 | 23, 25 | orbi12d 918 | . . . . . . 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 3698 | . . . . . 6 ⊢ ({𝑦} ∈ 𝑆 ↔ ({𝑦} ∈ 𝒫 𝐴 ∧ ({𝑦} ≼ ω ∨ (𝐴 ∖ {𝑦}) ≼ ω))) |
29 | 22, 28 | sylibr 234 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) → {𝑦} ∈ 𝑆) |
30 | 29 | rgen 3061 | . . . 4 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ 𝑆 |
31 | eqid 2735 | . . . . 5 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) = (𝑦 ∈ (0[,]1) ↦ {𝑦}) | |
32 | 31 | rnmptss 7143 | . . . 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 4030 | . 2 ⊢ 𝑋 ⊆ 𝑆 |
35 | ovex 7464 | . . . . . 6 ⊢ (0[,]2) ∈ V | |
36 | 12, 35 | eqeltri 2835 | . . . . 5 ⊢ 𝐴 ∈ V |
37 | 36 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ∈ V) |
38 | 37, 27 | salexct 46290 | . . 3 ⊢ (⊤ → 𝑆 ∈ SAlg) |
39 | 38 | mptru 1544 | . 2 ⊢ 𝑆 ∈ SAlg |
40 | ovex 7464 | . . . . . . . . 9 ⊢ (0[,]1) ∈ V | |
41 | 40 | mptex 7243 | . . . . . . . 8 ⊢ (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
42 | 41 | rnex 7933 | . . . . . . 7 ⊢ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) ∈ V |
43 | 1, 42 | eqeltri 2835 | . . . . . 6 ⊢ 𝑋 ∈ V |
44 | 43 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝑋 ∈ V) |
45 | salgensscntex.g | . . . . 5 ⊢ 𝐺 = (SalGen‘𝑋) | |
46 | 1 | unieqi 4924 | . . . . . 6 ⊢ ∪ 𝑋 = ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) |
47 | vsnex 5440 | . . . . . . . . 9 ⊢ {𝑦} ∈ V | |
48 | 47 | rgenw 3063 | . . . . . . . 8 ⊢ ∀𝑦 ∈ (0[,]1){𝑦} ∈ V |
49 | dfiun3g 5981 | . . . . . . . 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 2744 | . . . . . 6 ⊢ ∪ ran (𝑦 ∈ (0[,]1) ↦ {𝑦}) = ∪ 𝑦 ∈ (0[,]1){𝑦} |
52 | iunid 5065 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (0[,]1){𝑦} = (0[,]1) | |
53 | 46, 51, 52 | 3eqtrri 2768 | . . . . 5 ⊢ (0[,]1) = ∪ 𝑋 |
54 | 44, 45, 53 | unisalgen 46296 | . . . 4 ⊢ (⊤ → (0[,]1) ∈ 𝐺) |
55 | 54 | mptru 1544 | . . 3 ⊢ (0[,]1) ∈ 𝐺 |
56 | eqid 2735 | . . . 4 ⊢ (0[,]1) = (0[,]1) | |
57 | 12, 27, 56 | salexct2 46295 | . . 3 ⊢ ¬ (0[,]1) ∈ 𝑆 |
58 | nelss 4061 | . . 3 ⊢ (((0[,]1) ∈ 𝐺 ∧ ¬ (0[,]1) ∈ 𝑆) → ¬ 𝐺 ⊆ 𝑆) | |
59 | 55, 57, 58 | mp2an 692 | . 2 ⊢ ¬ 𝐺 ⊆ 𝑆 |
60 | 34, 39, 59 | 3pm3.2i 1338 | 1 ⊢ (𝑋 ⊆ 𝑆 ∧ 𝑆 ∈ SAlg ∧ ¬ 𝐺 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 ∪ cuni 4912 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 Fincfn 8984 ℝcr 11152 0cc0 11153 1c1 11154 ≤ cle 11294 2c2 12319 [,]cicc 13387 SAlgcsalg 46264 SalGencsalgen 46268 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-ntr 23044 df-salg 46265 df-salgen 46269 |
This theorem is referenced by: (None) |
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