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 non trivial sigma-algebra, see salexct 39228. (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 9965 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 ∈ ℝ*) |
3 | 1re 9918 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
4 | 3 | rexri 9976 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℝ*) |
6 | 0lt1 10429 | . . . . . . . 8 ⊢ 0 < 1 | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 < 1) |
8 | salexct2.3 | . . . . . . 7 ⊢ 𝐵 = (0[,]1) | |
9 | 2, 5, 7, 8 | iccnct 38615 | . . . . . 6 ⊢ (⊤ → ¬ 𝐵 ≼ ω) |
10 | 9 | trud 1484 | . . . . 5 ⊢ ¬ 𝐵 ≼ ω |
11 | 2re 10967 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
12 | 11 | rexri 9976 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
13 | 12 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 2 ∈ ℝ*) |
14 | 1lt2 11071 | . . . . . . . . 9 ⊢ 1 < 2 | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 1 < 2) |
16 | eqid 2610 | . . . . . . . 8 ⊢ (1(,]2) = (1(,]2) | |
17 | 5, 13, 15, 16 | iocnct 38614 | . . . . . . 7 ⊢ (⊤ → ¬ (1(,]2) ≼ ω) |
18 | 17 | trud 1484 | . . . . . 6 ⊢ ¬ (1(,]2) ≼ ω |
19 | salexct2.1 | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
20 | 19, 8 | difeq12i 3688 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐵) = ((0[,]2) ∖ (0[,]1)) |
21 | 2, 5, 7 | xrltled 38427 | . . . . . . . . . 10 ⊢ (⊤ → 0 ≤ 1) |
22 | 2, 5, 13, 21 | iccdificc 38613 | . . . . . . . . 9 ⊢ (⊤ → ((0[,]2) ∖ (0[,]1)) = (1(,]2)) |
23 | 22 | trud 1484 | . . . . . . . 8 ⊢ ((0[,]2) ∖ (0[,]1)) = (1(,]2) |
24 | 20, 23 | eqtri 2632 | . . . . . . 7 ⊢ (𝐴 ∖ 𝐵) = (1(,]2) |
25 | 24 | breq1i 4590 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ≼ ω ↔ (1(,]2) ≼ ω) |
26 | 18, 25 | mtbir 312 | . . . . 5 ⊢ ¬ (𝐴 ∖ 𝐵) ≼ ω |
27 | 10, 26 | pm3.2i 470 | . . . 4 ⊢ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω) |
28 | ioran 510 | . . . 4 ⊢ (¬ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω) ↔ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω)) | |
29 | 27, 28 | mpbir 220 | . . 3 ⊢ ¬ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω) |
30 | 29 | intnan 951 | . 2 ⊢ ¬ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω)) |
31 | breq1 4586 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω)) | |
32 | difeq2 3684 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵)) | |
33 | 32 | breq1d 4593 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐴 ∖ 𝑥) ≼ ω ↔ (𝐴 ∖ 𝐵) ≼ ω)) |
34 | 31, 33 | orbi12d 742 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω) ↔ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω))) |
35 | salexct2.2 | . . 3 ⊢ 𝑆 = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑥 ≼ ω ∨ (𝐴 ∖ 𝑥) ≼ ω)} | |
36 | 34, 35 | elrab2 3333 | . 2 ⊢ (𝐵 ∈ 𝑆 ↔ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω))) |
37 | 30, 36 | mtbir 312 | 1 ⊢ ¬ 𝐵 ∈ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 382 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 𝒫 cpw 4108 class class class wbr 4583 (class class class)co 6549 ωcom 6957 ≼ cdom 7839 0cc0 9815 1c1 9816 ℝ*cxr 9952 < clt 9953 2c2 10947 (,]cioc 12047 [,]cicc 12049 |
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-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 |
This theorem is referenced by: salexct3 39236 salgencntex 39237 salgensscntex 39238 |
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