Step | Hyp | Ref
| Expression |
1 | | elgrug 10285 |
. . . . . . 7
⊢ (𝑈 ∈ Univ → (𝑈 ∈ Univ ↔ (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)))) |
2 | 1 | ibi 270 |
. . . . . 6
⊢ (𝑈 ∈ Univ → (Tr 𝑈 ∧ ∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈))) |
3 | 2 | simprd 499 |
. . . . 5
⊢ (𝑈 ∈ Univ →
∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈)) |
4 | | preq2 4622 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐵 → {𝑥, 𝑦} = {𝑥, 𝐵}) |
5 | 4 | eleq1d 2817 |
. . . . . . . . 9
⊢ (𝑦 = 𝐵 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝑥, 𝐵} ∈ 𝑈)) |
6 | 5 | rspccv 3521 |
. . . . . . . 8
⊢
(∀𝑦 ∈
𝑈 {𝑥, 𝑦} ∈ 𝑈 → (𝐵 ∈ 𝑈 → {𝑥, 𝐵} ∈ 𝑈)) |
7 | 6 | 3ad2ant2 1135 |
. . . . . . 7
⊢
((𝒫 𝑥 ∈
𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → (𝐵 ∈ 𝑈 → {𝑥, 𝐵} ∈ 𝑈)) |
8 | 7 | com12 32 |
. . . . . 6
⊢ (𝐵 ∈ 𝑈 → ((𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → {𝑥, 𝐵} ∈ 𝑈)) |
9 | 8 | ralimdv 3092 |
. . . . 5
⊢ (𝐵 ∈ 𝑈 → (∀𝑥 ∈ 𝑈 (𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈 ∧ ∀𝑦 ∈ (𝑈 ↑m 𝑥)∪ ran 𝑦 ∈ 𝑈) → ∀𝑥 ∈ 𝑈 {𝑥, 𝐵} ∈ 𝑈)) |
10 | 3, 9 | syl5com 31 |
. . . 4
⊢ (𝑈 ∈ Univ → (𝐵 ∈ 𝑈 → ∀𝑥 ∈ 𝑈 {𝑥, 𝐵} ∈ 𝑈)) |
11 | | preq1 4621 |
. . . . . 6
⊢ (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵}) |
12 | 11 | eleq1d 2817 |
. . . . 5
⊢ (𝑥 = 𝐴 → ({𝑥, 𝐵} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈)) |
13 | 12 | rspccv 3521 |
. . . 4
⊢
(∀𝑥 ∈
𝑈 {𝑥, 𝐵} ∈ 𝑈 → (𝐴 ∈ 𝑈 → {𝐴, 𝐵} ∈ 𝑈)) |
14 | 10, 13 | syl6 35 |
. . 3
⊢ (𝑈 ∈ Univ → (𝐵 ∈ 𝑈 → (𝐴 ∈ 𝑈 → {𝐴, 𝐵} ∈ 𝑈))) |
15 | 14 | com23 86 |
. 2
⊢ (𝑈 ∈ Univ → (𝐴 ∈ 𝑈 → (𝐵 ∈ 𝑈 → {𝐴, 𝐵} ∈ 𝑈))) |
16 | 15 | 3imp 1112 |
1
⊢ ((𝑈 ∈ Univ ∧ 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈) |