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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issalnnd | Structured version Visualization version GIF version |
Description: Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
issalnnd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
issalnnd.z | ⊢ (𝜑 → ∅ ∈ 𝑆) |
issalnnd.x | ⊢ 𝑋 = ∪ 𝑆 |
issalnnd.d | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆) |
issalnnd.i | ⊢ ((𝜑 ∧ 𝑒:ℕ⟶𝑆) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆) |
Ref | Expression |
---|---|
issalnnd | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issalnnd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
2 | issalnnd.z | . 2 ⊢ (𝜑 → ∅ ∈ 𝑆) | |
3 | issalnnd.x | . 2 ⊢ 𝑋 = ∪ 𝑆 | |
4 | issalnnd.d | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆) | |
5 | unieq 4919 | . . . . . . 7 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∪ ∅) | |
6 | uni0 4938 | . . . . . . . 8 ⊢ ∪ ∅ = ∅ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑦 = ∅ → ∪ ∅ = ∅) |
8 | 5, 7 | eqtrd 2768 | . . . . . 6 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∅) |
9 | 8 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∪ 𝑦 = ∅) |
10 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∅ ∈ 𝑆) |
11 | 9, 10 | eqeltrd 2829 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
12 | 11 | 3ad2antl1 1183 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
13 | neqne 2945 | . . . . 5 ⊢ (¬ 𝑦 = ∅ → 𝑦 ≠ ∅) | |
14 | 13 | adantl 481 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅) |
15 | nnfoctb 44411 | . . . . . 6 ⊢ ((𝑦 ≼ ω ∧ 𝑦 ≠ ∅) → ∃𝑒 𝑒:ℕ–onto→𝑦) | |
16 | 15 | 3ad2antl3 1185 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → ∃𝑒 𝑒:ℕ–onto→𝑦) |
17 | founiiun 44552 | . . . . . . . . . . 11 ⊢ (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) | |
18 | 17 | adantl 481 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) |
19 | simpll 766 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → 𝜑) | |
20 | fof 6811 | . . . . . . . . . . . . . 14 ⊢ (𝑒:ℕ–onto→𝑦 → 𝑒:ℕ⟶𝑦) | |
21 | 20 | adantl 481 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑒:ℕ⟶𝑦) |
22 | elpwi 4610 | . . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 𝑆 → 𝑦 ⊆ 𝑆) | |
23 | 22 | adantr 480 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑦 ⊆ 𝑆) |
24 | 21, 23 | fssd 6740 | . . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑒:ℕ⟶𝑆) |
25 | 24 | adantll 713 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → 𝑒:ℕ⟶𝑆) |
26 | issalnnd.i | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒:ℕ⟶𝑆) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆) | |
27 | 19, 25, 26 | syl2anc 583 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆) |
28 | 18, 27 | eqeltrd 2829 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑦 ∈ 𝑆) |
29 | 28 | ex 412 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
30 | 29 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑦 ≠ ∅) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
31 | 30 | 3adantl3 1166 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
32 | 31 | exlimdv 1929 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → (∃𝑒 𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
33 | 16, 32 | mpd 15 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → ∪ 𝑦 ∈ 𝑆) |
34 | 14, 33 | syldan 590 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ ¬ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
35 | 12, 34 | pm2.61dan 812 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) |
36 | 1, 2, 3, 4, 35 | issald 45721 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4908 ∪ ciun 4996 class class class wbr 5148 ⟶wf 6544 –onto→wfo 6546 ‘cfv 6548 ωcom 7870 ≼ cdom 8961 ℕcn 12242 SAlgcsalg 45696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-salg 45697 |
This theorem is referenced by: dmvolsal 45734 subsalsal 45747 smfresal 46176 |
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