Step | Hyp | Ref
| Expression |
1 | | tcvalg 9496 |
. . . . 5
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)}) |
2 | | fvex 6787 |
. . . . 5
⊢
(TC‘𝐴) ∈
V |
3 | 1, 2 | eqeltrrdi 2848 |
. . . 4
⊢ (𝐴 ∈ 𝑉 → ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) |
4 | | intexab 5263 |
. . . 4
⊢
(∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) |
5 | 3, 4 | sylibr 233 |
. . 3
⊢ (𝐴 ∈ 𝑉 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥)) |
6 | | ssin 4164 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵)) |
7 | 6 | biimpi 215 |
. . . . . . . 8
⊢ ((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑥 ∩ 𝐵)) |
8 | | trin 5201 |
. . . . . . . 8
⊢ ((Tr
𝑥 ∧ Tr 𝐵) → Tr (𝑥 ∩ 𝐵)) |
9 | 7, 8 | anim12i 613 |
. . . . . . 7
⊢ (((𝐴 ⊆ 𝑥 ∧ 𝐴 ⊆ 𝐵) ∧ (Tr 𝑥 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))) |
10 | 9 | an4s 657 |
. . . . . 6
⊢ (((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) ∧ (𝐴 ⊆ 𝐵 ∧ Tr 𝐵)) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))) |
11 | 10 | expcom 414 |
. . . . 5
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))) |
12 | | vex 3436 |
. . . . . . . . 9
⊢ 𝑥 ∈ V |
13 | 12 | inex1 5241 |
. . . . . . . 8
⊢ (𝑥 ∩ 𝐵) ∈ V |
14 | | sseq2 3947 |
. . . . . . . . 9
⊢ (𝑦 = (𝑥 ∩ 𝐵) → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ (𝑥 ∩ 𝐵))) |
15 | | treq 5197 |
. . . . . . . . 9
⊢ (𝑦 = (𝑥 ∩ 𝐵) → (Tr 𝑦 ↔ Tr (𝑥 ∩ 𝐵))) |
16 | 14, 15 | anbi12d 631 |
. . . . . . . 8
⊢ (𝑦 = (𝑥 ∩ 𝐵) → ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)))) |
17 | 13, 16 | elab 3609 |
. . . . . . 7
⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ↔ (𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵))) |
18 | | intss1 4894 |
. . . . . . 7
⊢ ((𝑥 ∩ 𝐵) ∈ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵)) |
19 | 17, 18 | sylbir 234 |
. . . . . 6
⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩
{𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ (𝑥 ∩ 𝐵)) |
20 | | inss2 4163 |
. . . . . 6
⊢ (𝑥 ∩ 𝐵) ⊆ 𝐵 |
21 | 19, 20 | sstrdi 3933 |
. . . . 5
⊢ ((𝐴 ⊆ (𝑥 ∩ 𝐵) ∧ Tr (𝑥 ∩ 𝐵)) → ∩
{𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵) |
22 | 11, 21 | syl6 35 |
. . . 4
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) |
23 | 22 | exlimdv 1936 |
. . 3
⊢ ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) |
24 | 5, 23 | syl5com 31 |
. 2
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) |
25 | | tcvalg 9496 |
. . 3
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)}) |
26 | 25 | sseq1d 3952 |
. 2
⊢ (𝐴 ∈ 𝑉 → ((TC‘𝐴) ⊆ 𝐵 ↔ ∩ {𝑦 ∣ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)} ⊆ 𝐵)) |
27 | 24, 26 | sylibrd 258 |
1
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵)) |