Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → 𝑦 ∈ ∪ 𝐴 ) |
2 |
1
|
a1i |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → 𝑦 ∈ ∪ 𝐴 ) ) |
3 |
|
eluni |
⊢ ( 𝑦 ∈ ∪ 𝐴 ↔ ∃ 𝑞 ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) ) |
4 |
2 3
|
syl6ib |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ∃ 𝑞 ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) ) ) |
5 |
|
simpl |
⊢ ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → 𝑧 ∈ 𝑦 ) |
6 |
5
|
a1i |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → 𝑧 ∈ 𝑦 ) ) |
7 |
|
simpl |
⊢ ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑦 ∈ 𝑞 ) |
8 |
7
|
2a1i |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑦 ∈ 𝑞 ) ) ) |
9 |
|
simpr |
⊢ ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑞 ∈ 𝐴 ) |
10 |
9
|
2a1i |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑞 ∈ 𝐴 ) ) ) |
11 |
|
rspsbc |
⊢ ( 𝑞 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → [ 𝑞 / 𝑥 ] Tr 𝑥 ) ) |
12 |
11
|
com12 |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( 𝑞 ∈ 𝐴 → [ 𝑞 / 𝑥 ] Tr 𝑥 ) ) |
13 |
10 12
|
syl6d |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → [ 𝑞 / 𝑥 ] Tr 𝑥 ) ) ) |
14 |
|
trsbc |
⊢ ( 𝑞 ∈ 𝐴 → ( [ 𝑞 / 𝑥 ] Tr 𝑥 ↔ Tr 𝑞 ) ) |
15 |
14
|
biimpd |
⊢ ( 𝑞 ∈ 𝐴 → ( [ 𝑞 / 𝑥 ] Tr 𝑥 → Tr 𝑞 ) ) |
16 |
10 13 15
|
ee33 |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → Tr 𝑞 ) ) ) |
17 |
|
trel |
⊢ ( Tr 𝑞 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞 ) → 𝑧 ∈ 𝑞 ) ) |
18 |
17
|
expdcom |
⊢ ( 𝑧 ∈ 𝑦 → ( 𝑦 ∈ 𝑞 → ( Tr 𝑞 → 𝑧 ∈ 𝑞 ) ) ) |
19 |
6 8 16 18
|
ee233 |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑧 ∈ 𝑞 ) ) ) |
20 |
|
elunii |
⊢ ( ( 𝑧 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) |
21 |
20
|
ex |
⊢ ( 𝑧 ∈ 𝑞 → ( 𝑞 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴 ) ) |
22 |
19 10 21
|
ee33 |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ) ) |
23 |
22
|
alrimdv |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ∀ 𝑞 ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ) ) |
24 |
|
19.23v |
⊢ ( ∀ 𝑞 ( ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ↔ ( ∃ 𝑞 ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ) |
25 |
23 24
|
syl6ib |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → ( ∃ 𝑞 ( 𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ) ) |
26 |
4 25
|
mpdd |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ) |
27 |
26
|
alrimivv |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → ∀ 𝑧 ∀ 𝑦 ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ) |
28 |
|
dftr2 |
⊢ ( Tr ∪ 𝐴 ↔ ∀ 𝑧 ∀ 𝑦 ( ( 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴 ) → 𝑧 ∈ ∪ 𝐴 ) ) |
29 |
27 28
|
sylibr |
⊢ ( ∀ 𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴 ) |