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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 44661. (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 11207 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 ∈ ℝ*) |
3 | 1xr 11219 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℝ*) |
5 | 0lt1 11682 | . . . . . . . 8 ⊢ 0 < 1 | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 < 1) |
7 | salexct2.3 | . . . . . . 7 ⊢ 𝐵 = (0[,]1) | |
8 | 2, 4, 6, 7 | iccnct 43865 | . . . . . 6 ⊢ (⊤ → ¬ 𝐵 ≼ ω) |
9 | 8 | mptru 1549 | . . . . 5 ⊢ ¬ 𝐵 ≼ ω |
10 | 2re 12232 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
11 | 10 | rexri 11218 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 2 ∈ ℝ*) |
13 | 1lt2 12329 | . . . . . . . . 9 ⊢ 1 < 2 | |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 1 < 2) |
15 | eqid 2733 | . . . . . . . 8 ⊢ (1(,]2) = (1(,]2) | |
16 | 4, 12, 14, 15 | iocnct 43864 | . . . . . . 7 ⊢ (⊤ → ¬ (1(,]2) ≼ ω) |
17 | 16 | mptru 1549 | . . . . . 6 ⊢ ¬ (1(,]2) ≼ ω |
18 | salexct2.1 | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
19 | 18, 7 | difeq12i 4081 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐵) = ((0[,]2) ∖ (0[,]1)) |
20 | 2, 4, 6 | xrltled 13075 | . . . . . . . . . 10 ⊢ (⊤ → 0 ≤ 1) |
21 | 2, 4, 12, 20 | iccdificc 43863 | . . . . . . . . 9 ⊢ (⊤ → ((0[,]2) ∖ (0[,]1)) = (1(,]2)) |
22 | 21 | mptru 1549 | . . . . . . . 8 ⊢ ((0[,]2) ∖ (0[,]1)) = (1(,]2) |
23 | 19, 22 | eqtri 2761 | . . . . . . 7 ⊢ (𝐴 ∖ 𝐵) = (1(,]2) |
24 | 23 | breq1i 5113 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ≼ ω ↔ (1(,]2) ≼ ω) |
25 | 17, 24 | mtbir 323 | . . . . 5 ⊢ ¬ (𝐴 ∖ 𝐵) ≼ ω |
26 | 9, 25 | pm3.2i 472 | . . . 4 ⊢ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω) |
27 | ioran 983 | . . . 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 488 | . 2 ⊢ ¬ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω)) |
30 | breq1 5109 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω)) | |
31 | difeq2 4077 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵)) | |
32 | 31 | breq1d 5116 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝐵) ≼ ω)) |
33 | 30, 32 | orbi12d 918 | . . 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 3649 | . 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 397 ∨ wo 846 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 {crab 3406 ∖ cdif 3908 𝒫 cpw 4561 class class class wbr 5106 (class class class)co 7358 ωcom 7803 ≼ cdom 8884 0cc0 11056 1c1 11057 ℝ*cxr 11193 < clt 11194 2c2 12213 (,]cioc 13271 [,]cicc 13273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-oadd 8417 df-omul 8418 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-acn 9883 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13274 df-ioc 13275 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-limsup 15359 df-clim 15376 df-rlim 15377 df-sum 15577 df-topgen 17330 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-top 22259 df-topon 22276 df-bases 22312 df-ntr 22387 |
This theorem is referenced by: salexct3 44669 salgencntex 44670 salgensscntex 44671 |
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