Proof of Theorem isros
Step | Hyp | Ref
| Expression |
1 | | eleq2 2829 |
. . . 4
⊢ (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆)) |
2 | | eleq2 2829 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → ((𝑥 ∪ 𝑦) ∈ 𝑠 ↔ (𝑥 ∪ 𝑦) ∈ 𝑆)) |
3 | | eleq2 2829 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → ((𝑥 ∖ 𝑦) ∈ 𝑠 ↔ (𝑥 ∖ 𝑦) ∈ 𝑆)) |
4 | 2, 3 | anbi12d 631 |
. . . . . 6
⊢ (𝑠 = 𝑆 → (((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆))) |
5 | 4 | raleqbi1dv 3339 |
. . . . 5
⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆))) |
6 | 5 | raleqbi1dv 3339 |
. . . 4
⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆))) |
7 | 1, 6 | anbi12d 631 |
. . 3
⊢ (𝑠 = 𝑆 → ((∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)))) |
8 | | isros.1 |
. . 3
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} |
9 | 7, 8 | elrab2 3629 |
. 2
⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)))) |
10 | | 3anass 1094 |
. 2
⊢ ((𝑆 ∈ 𝒫 𝒫
𝑂 ∧ ∅ ∈
𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)))) |
11 | | uneq1 4095 |
. . . . . 6
⊢ (𝑥 = 𝑢 → (𝑥 ∪ 𝑦) = (𝑢 ∪ 𝑦)) |
12 | 11 | eleq1d 2825 |
. . . . 5
⊢ (𝑥 = 𝑢 → ((𝑥 ∪ 𝑦) ∈ 𝑆 ↔ (𝑢 ∪ 𝑦) ∈ 𝑆)) |
13 | | difeq1 4055 |
. . . . . 6
⊢ (𝑥 = 𝑢 → (𝑥 ∖ 𝑦) = (𝑢 ∖ 𝑦)) |
14 | 13 | eleq1d 2825 |
. . . . 5
⊢ (𝑥 = 𝑢 → ((𝑥 ∖ 𝑦) ∈ 𝑆 ↔ (𝑢 ∖ 𝑦) ∈ 𝑆)) |
15 | 12, 14 | anbi12d 631 |
. . . 4
⊢ (𝑥 = 𝑢 → (((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆) ↔ ((𝑢 ∪ 𝑦) ∈ 𝑆 ∧ (𝑢 ∖ 𝑦) ∈ 𝑆))) |
16 | | uneq2 4096 |
. . . . . 6
⊢ (𝑦 = 𝑣 → (𝑢 ∪ 𝑦) = (𝑢 ∪ 𝑣)) |
17 | 16 | eleq1d 2825 |
. . . . 5
⊢ (𝑦 = 𝑣 → ((𝑢 ∪ 𝑦) ∈ 𝑆 ↔ (𝑢 ∪ 𝑣) ∈ 𝑆)) |
18 | | difeq2 4056 |
. . . . . 6
⊢ (𝑦 = 𝑣 → (𝑢 ∖ 𝑦) = (𝑢 ∖ 𝑣)) |
19 | 18 | eleq1d 2825 |
. . . . 5
⊢ (𝑦 = 𝑣 → ((𝑢 ∖ 𝑦) ∈ 𝑆 ↔ (𝑢 ∖ 𝑣) ∈ 𝑆)) |
20 | 17, 19 | anbi12d 631 |
. . . 4
⊢ (𝑦 = 𝑣 → (((𝑢 ∪ 𝑦) ∈ 𝑆 ∧ (𝑢 ∖ 𝑦) ∈ 𝑆) ↔ ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))) |
21 | 15, 20 | cbvral2vw 3394 |
. . 3
⊢
(∀𝑥 ∈
𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆) ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)) |
22 | 21 | 3anbi3i 1158 |
. 2
⊢ ((𝑆 ∈ 𝒫 𝒫
𝑂 ∧ ∅ ∈
𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))) |
23 | 9, 10, 22 | 3bitr2i 299 |
1
⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))) |