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 4839 | . . . . . . 7 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∪ ∅) | |
6 | uni0 4858 | . . . . . . . 8 ⊢ ∪ ∅ = ∅ | |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝑦 = ∅ → ∪ ∅ = ∅) |
8 | 5, 7 | eqtrd 2856 | . . . . . 6 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∅) |
9 | 8 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∪ 𝑦 = ∅) |
10 | 2 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∅ ∈ 𝑆) |
11 | 9, 10 | eqeltrd 2913 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
12 | 11 | 3ad2antl1 1181 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
13 | neqne 3024 | . . . . 5 ⊢ (¬ 𝑦 = ∅ → 𝑦 ≠ ∅) | |
14 | 13 | adantl 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅) |
15 | nnfoctb 41302 | . . . . . 6 ⊢ ((𝑦 ≼ ω ∧ 𝑦 ≠ ∅) → ∃𝑒 𝑒:ℕ–onto→𝑦) | |
16 | 15 | 3ad2antl3 1183 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → ∃𝑒 𝑒:ℕ–onto→𝑦) |
17 | founiiun 41428 | . . . . . . . . . . 11 ⊢ (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) | |
18 | 17 | adantl 484 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ (𝑒‘𝑛)) |
19 | simpll 765 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → 𝜑) | |
20 | fof 6584 | . . . . . . . . . . . . . 14 ⊢ (𝑒:ℕ–onto→𝑦 → 𝑒:ℕ⟶𝑦) | |
21 | 20 | adantl 484 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑒:ℕ⟶𝑦) |
22 | elpwi 4550 | . . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝒫 𝑆 → 𝑦 ⊆ 𝑆) | |
23 | 22 | adantr 483 | . . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑦 ⊆ 𝑆) |
24 | 21, 23 | fssd 6522 | . . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑒:ℕ–onto→𝑦) → 𝑒:ℕ⟶𝑆) |
25 | 24 | adantll 712 | . . . . . . . . . . 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 586 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆) |
28 | 18, 27 | eqeltrd 2913 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑒:ℕ–onto→𝑦) → ∪ 𝑦 ∈ 𝑆) |
29 | 28 | ex 415 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
30 | 29 | adantr 483 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑦 ≠ ∅) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
31 | 30 | 3adantl3 1164 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ 𝑦 ≠ ∅) → (𝑒:ℕ–onto→𝑦 → ∪ 𝑦 ∈ 𝑆)) |
32 | 31 | exlimdv 1930 | . . . . 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 593 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) ∧ ¬ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝑆) |
35 | 12, 34 | pm2.61dan 811 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆) |
36 | 1, 2, 3, 4, 35 | issald 42610 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4831 ∪ ciun 4911 class class class wbr 5058 ⟶wf 6345 –onto→wfo 6347 ‘cfv 6349 ωcom 7574 ≼ cdom 8501 ℕcn 11632 SAlgcsalg 42587 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-salg 42588 |
This theorem is referenced by: dmvolsal 42623 subsalsal 42636 smfresal 43057 |
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