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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 46290. (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 11306 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 ∈ ℝ*) |
3 | 1xr 11318 | . . . . . . . 8 ⊢ 1 ∈ ℝ* | |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℝ*) |
5 | 0lt1 11783 | . . . . . . . 8 ⊢ 0 < 1 | |
6 | 5 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 0 < 1) |
7 | salexct2.3 | . . . . . . 7 ⊢ 𝐵 = (0[,]1) | |
8 | 2, 4, 6, 7 | iccnct 45494 | . . . . . 6 ⊢ (⊤ → ¬ 𝐵 ≼ ω) |
9 | 8 | mptru 1544 | . . . . 5 ⊢ ¬ 𝐵 ≼ ω |
10 | 2re 12338 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
11 | 10 | rexri 11317 | . . . . . . . . 9 ⊢ 2 ∈ ℝ* |
12 | 11 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 2 ∈ ℝ*) |
13 | 1lt2 12435 | . . . . . . . . 9 ⊢ 1 < 2 | |
14 | 13 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → 1 < 2) |
15 | eqid 2735 | . . . . . . . 8 ⊢ (1(,]2) = (1(,]2) | |
16 | 4, 12, 14, 15 | iocnct 45493 | . . . . . . 7 ⊢ (⊤ → ¬ (1(,]2) ≼ ω) |
17 | 16 | mptru 1544 | . . . . . 6 ⊢ ¬ (1(,]2) ≼ ω |
18 | salexct2.1 | . . . . . . . . 9 ⊢ 𝐴 = (0[,]2) | |
19 | 18, 7 | difeq12i 4134 | . . . . . . . 8 ⊢ (𝐴 ∖ 𝐵) = ((0[,]2) ∖ (0[,]1)) |
20 | 2, 4, 6 | xrltled 13189 | . . . . . . . . . 10 ⊢ (⊤ → 0 ≤ 1) |
21 | 2, 4, 12, 20 | iccdificc 45492 | . . . . . . . . 9 ⊢ (⊤ → ((0[,]2) ∖ (0[,]1)) = (1(,]2)) |
22 | 21 | mptru 1544 | . . . . . . . 8 ⊢ ((0[,]2) ∖ (0[,]1)) = (1(,]2) |
23 | 19, 22 | eqtri 2763 | . . . . . . 7 ⊢ (𝐴 ∖ 𝐵) = (1(,]2) |
24 | 23 | breq1i 5155 | . . . . . 6 ⊢ ((𝐴 ∖ 𝐵) ≼ ω ↔ (1(,]2) ≼ ω) |
25 | 17, 24 | mtbir 323 | . . . . 5 ⊢ ¬ (𝐴 ∖ 𝐵) ≼ ω |
26 | 9, 25 | pm3.2i 470 | . . . 4 ⊢ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω) |
27 | ioran 985 | . . . 4 ⊢ (¬ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω) ↔ (¬ 𝐵 ≼ ω ∧ ¬ (𝐴 ∖ 𝐵) ≼ ω)) | |
28 | 26, 27 | mpbir 231 | . . 3 ⊢ ¬ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω) |
29 | 28 | intnan 486 | . 2 ⊢ ¬ (𝐵 ∈ 𝒫 𝐴 ∧ (𝐵 ≼ ω ∨ (𝐴 ∖ 𝐵) ≼ ω)) |
30 | breq1 5151 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ≼ ω ↔ 𝐵 ≼ ω)) | |
31 | difeq2 4130 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐵)) | |
32 | 31 | breq1d 5158 | . . . 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 3698 | . 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 395 ∨ wo 847 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 {crab 3433 ∖ cdif 3960 𝒫 cpw 4605 class class class wbr 5148 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 0cc0 11153 1c1 11154 ℝ*cxr 11292 < clt 11293 2c2 12319 (,]cioc 13385 [,]cicc 13387 |
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-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 |
This theorem is referenced by: salexct3 46298 salgencntex 46299 salgensscntex 46300 |
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