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 4830 | . . . . . . 7 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∪ ∅) | |
6 | uni0 4849 | . . . . . . . 8 ⊢ ∪ ∅ = ∅ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑦 = ∅ → ∪ ∅ = ∅) |
8 | 5, 7 | eqtrd 2777 | . . . . . 6 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∅) |
9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∪ 𝑦 = ∅) |
10 | 2 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∅ ∈ 𝑆) |
11 | 9, 10 | eqeltrd 2838 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
12 | 11 | 3ad2antl1 1187 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
13 | neqne 2948 | . . . . 5 ⊢ (¬ 𝑦 = ∅ → 𝑦 ≠ ∅) | |
14 | 13 | adantl 485 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅) |
15 | nnfoctb 42268 | . . . . . 6 ⊢ ((𝑦 ≼ ω ∧ 𝑦 ≠ ∅) → ∃𝑒 𝑒:ℕ–onto→𝑦) | |
16 | 15 | 3ad2antl3 1189 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → ∃𝑒 𝑒:ℕ–onto→𝑦) |
17 | founiiun 42388 | . . . . . . . . . . 11 ⊢ (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) | |
18 | 17 | adantl 485 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) |
19 | simpll 767 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → 𝜑) | |
20 | fof 6633 | . . . . . . . . . . . . . 14 ⊢ (𝑒:ℕ–onto→𝑦 → 𝑒:ℕ⟶𝑦) | |
21 | 20 | adantl 485 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑒:ℕ⟶𝑦) |
22 | elpwi 4522 | . . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 𝑆 → 𝑦 ⊆ 𝑆) | |
23 | 22 | adantr 484 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑦 ⊆ 𝑆) |
24 | 21, 23 | fssd 6563 | . . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑒:ℕ⟶𝑆) |
25 | 24 | adantll 714 | . . . . . . . . . . 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 587 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆) |
28 | 18, 27 | eqeltrd 2838 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑦 ∈ 𝑆) |
29 | 28 | ex 416 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
30 | 29 | adantr 484 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑦 ≠ ∅) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
31 | 30 | 3adantl3 1170 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
32 | 31 | exlimdv 1941 | . . . . 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 594 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ ¬ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
35 | 12, 34 | pm2.61dan 813 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) |
36 | 1, 2, 3, 4, 35 | issald 43547 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ≠ wne 2940 ∖ cdif 3863 ⊆ wss 3866 ∅c0 4237 𝒫 cpw 4513 ∪ cuni 4819 ∪ ciun 4904 class class class wbr 5053 ⟶wf 6376 –onto→wfo 6378 ‘cfv 6380 ωcom 7644 ≼ cdom 8624 ℕcn 11830 SAlgcsalg 43524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-salg 43525 |
This theorem is referenced by: dmvolsal 43560 subsalsal 43573 smfresal 43994 |
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