Step |
Hyp |
Ref |
Expression |
1 |
|
eliuniin2.1 |
⊢ Ⅎ 𝑥 𝐶 |
2 |
|
eliuniin2.2 |
⊢ 𝐴 = ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 |
3 |
2
|
eleq2i |
⊢ ( 𝑍 ∈ 𝐴 ↔ 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 ) |
4 |
|
eliun |
⊢ ( 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∃ 𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ) |
5 |
3 4
|
sylbb |
⊢ ( 𝑍 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ) |
6 |
|
eliin |
⊢ ( 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ( 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) ) |
7 |
6
|
ibi |
⊢ ( 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) |
8 |
7
|
a1i |
⊢ ( 𝑍 ∈ 𝐴 → ( 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) ) |
9 |
8
|
reximdv |
⊢ ( 𝑍 ∈ 𝐴 → ( ∃ 𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) ) |
10 |
5 9
|
mpd |
⊢ ( 𝑍 ∈ 𝐴 → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) |
11 |
|
nfcv |
⊢ Ⅎ 𝑥 ∅ |
12 |
1 11
|
nfne |
⊢ Ⅎ 𝑥 𝐶 ≠ ∅ |
13 |
|
nfv |
⊢ Ⅎ 𝑥 𝑍 ∈ 𝐴 |
14 |
|
simp2 |
⊢ ( ( 𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) → 𝑥 ∈ 𝐵 ) |
15 |
|
eliin2 |
⊢ ( 𝐶 ≠ ∅ → ( 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ↔ ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) ) |
16 |
15
|
biimpar |
⊢ ( ( 𝐶 ≠ ∅ ∧ ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) → 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ) |
17 |
|
rspe |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ) → ∃ 𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ) |
18 |
14 16 17
|
3imp3i2an |
⊢ ( ( 𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) → ∃ 𝑥 ∈ 𝐵 𝑍 ∈ ∩ 𝑦 ∈ 𝐶 𝐷 ) |
19 |
18 4
|
sylibr |
⊢ ( ( 𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) → 𝑍 ∈ ∪ 𝑥 ∈ 𝐵 ∩ 𝑦 ∈ 𝐶 𝐷 ) |
20 |
19 3
|
sylibr |
⊢ ( ( 𝐶 ≠ ∅ ∧ 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) → 𝑍 ∈ 𝐴 ) |
21 |
20
|
3exp |
⊢ ( 𝐶 ≠ ∅ → ( 𝑥 ∈ 𝐵 → ( ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 → 𝑍 ∈ 𝐴 ) ) ) |
22 |
12 13 21
|
rexlimd |
⊢ ( 𝐶 ≠ ∅ → ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 → 𝑍 ∈ 𝐴 ) ) |
23 |
10 22
|
impbid2 |
⊢ ( 𝐶 ≠ ∅ → ( 𝑍 ∈ 𝐴 ↔ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐶 𝑍 ∈ 𝐷 ) ) |