Step |
Hyp |
Ref |
Expression |
1 |
|
issalnnd.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
2 |
|
issalnnd.z |
⊢ ( 𝜑 → ∅ ∈ 𝑆 ) |
3 |
|
issalnnd.x |
⊢ 𝑋 = ∪ 𝑆 |
4 |
|
issalnnd.d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑆 ) → ( 𝑋 ∖ 𝑦 ) ∈ 𝑆 ) |
5 |
|
issalnnd.i |
⊢ ( ( 𝜑 ∧ 𝑒 : ℕ ⟶ 𝑆 ) → ∪ 𝑛 ∈ ℕ ( 𝑒 ‘ 𝑛 ) ∈ 𝑆 ) |
6 |
|
unieq |
⊢ ( 𝑦 = ∅ → ∪ 𝑦 = ∪ ∅ ) |
7 |
|
uni0 |
⊢ ∪ ∅ = ∅ |
8 |
7
|
a1i |
⊢ ( 𝑦 = ∅ → ∪ ∅ = ∅ ) |
9 |
6 8
|
eqtrd |
⊢ ( 𝑦 = ∅ → ∪ 𝑦 = ∅ ) |
10 |
9
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 = ∅ ) → ∪ 𝑦 = ∅ ) |
11 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 = ∅ ) → ∅ ∈ 𝑆 ) |
12 |
10 11
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑦 = ∅ ) → ∪ 𝑦 ∈ 𝑆 ) |
13 |
12
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) ∧ 𝑦 = ∅ ) → ∪ 𝑦 ∈ 𝑆 ) |
14 |
|
neqne |
⊢ ( ¬ 𝑦 = ∅ → 𝑦 ≠ ∅ ) |
15 |
14
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) ∧ ¬ 𝑦 = ∅ ) → 𝑦 ≠ ∅ ) |
16 |
|
nnfoctb |
⊢ ( ( 𝑦 ≼ ω ∧ 𝑦 ≠ ∅ ) → ∃ 𝑒 𝑒 : ℕ –onto→ 𝑦 ) |
17 |
16
|
3ad2antl3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → ∃ 𝑒 𝑒 : ℕ –onto→ 𝑦 ) |
18 |
|
founiiun |
⊢ ( 𝑒 : ℕ –onto→ 𝑦 → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ ( 𝑒 ‘ 𝑛 ) ) |
19 |
18
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto→ 𝑦 ) → ∪ 𝑦 = ∪ 𝑛 ∈ ℕ ( 𝑒 ‘ 𝑛 ) ) |
20 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto→ 𝑦 ) → 𝜑 ) |
21 |
|
fof |
⊢ ( 𝑒 : ℕ –onto→ 𝑦 → 𝑒 : ℕ ⟶ 𝑦 ) |
22 |
21
|
adantl |
⊢ ( ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑒 : ℕ –onto→ 𝑦 ) → 𝑒 : ℕ ⟶ 𝑦 ) |
23 |
|
elpwi |
⊢ ( 𝑦 ∈ 𝒫 𝑆 → 𝑦 ⊆ 𝑆 ) |
24 |
23
|
adantr |
⊢ ( ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑒 : ℕ –onto→ 𝑦 ) → 𝑦 ⊆ 𝑆 ) |
25 |
22 24
|
fssd |
⊢ ( ( 𝑦 ∈ 𝒫 𝑆 ∧ 𝑒 : ℕ –onto→ 𝑦 ) → 𝑒 : ℕ ⟶ 𝑆 ) |
26 |
25
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto→ 𝑦 ) → 𝑒 : ℕ ⟶ 𝑆 ) |
27 |
20 26 5
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto→ 𝑦 ) → ∪ 𝑛 ∈ ℕ ( 𝑒 ‘ 𝑛 ) ∈ 𝑆 ) |
28 |
19 27
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto→ 𝑦 ) → ∪ 𝑦 ∈ 𝑆 ) |
29 |
28
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ) → ( 𝑒 : ℕ –onto→ 𝑦 → ∪ 𝑦 ∈ 𝑆 ) ) |
30 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑦 ≠ ∅ ) → ( 𝑒 : ℕ –onto→ 𝑦 → ∪ 𝑦 ∈ 𝑆 ) ) |
31 |
30
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → ( 𝑒 : ℕ –onto→ 𝑦 → ∪ 𝑦 ∈ 𝑆 ) ) |
32 |
31
|
exlimdv |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → ( ∃ 𝑒 𝑒 : ℕ –onto→ 𝑦 → ∪ 𝑦 ∈ 𝑆 ) ) |
33 |
17 32
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → ∪ 𝑦 ∈ 𝑆 ) |
34 |
15 33
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) ∧ ¬ 𝑦 = ∅ ) → ∪ 𝑦 ∈ 𝑆 ) |
35 |
13 34
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ) → ∪ 𝑦 ∈ 𝑆 ) |
36 |
1 2 3 4 35
|
issald |
⊢ ( 𝜑 → 𝑆 ∈ SAlg ) |