Step | Hyp | Ref
| Expression |
1 | | simp2 1136 |
. . 3
⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆) |
2 | | simp3 1137 |
. . 3
⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆) |
3 | | isros.1 |
. . . . . 6
⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))} |
4 | 3 | isros 32133 |
. . . . 5
⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))) |
5 | 4 | simp3bi 1146 |
. . . 4
⊢ (𝑆 ∈ 𝑄 → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)) |
6 | 5 | 3ad2ant1 1132 |
. . 3
⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)) |
7 | | uneq1 4091 |
. . . . . 6
⊢ (𝑢 = 𝐴 → (𝑢 ∪ 𝑣) = (𝐴 ∪ 𝑣)) |
8 | 7 | eleq1d 2823 |
. . . . 5
⊢ (𝑢 = 𝐴 → ((𝑢 ∪ 𝑣) ∈ 𝑆 ↔ (𝐴 ∪ 𝑣) ∈ 𝑆)) |
9 | | difeq1 4051 |
. . . . . 6
⊢ (𝑢 = 𝐴 → (𝑢 ∖ 𝑣) = (𝐴 ∖ 𝑣)) |
10 | 9 | eleq1d 2823 |
. . . . 5
⊢ (𝑢 = 𝐴 → ((𝑢 ∖ 𝑣) ∈ 𝑆 ↔ (𝐴 ∖ 𝑣) ∈ 𝑆)) |
11 | 8, 10 | anbi12d 631 |
. . . 4
⊢ (𝑢 = 𝐴 → (((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆) ↔ ((𝐴 ∪ 𝑣) ∈ 𝑆 ∧ (𝐴 ∖ 𝑣) ∈ 𝑆))) |
12 | | uneq2 4092 |
. . . . . 6
⊢ (𝑣 = 𝐵 → (𝐴 ∪ 𝑣) = (𝐴 ∪ 𝐵)) |
13 | 12 | eleq1d 2823 |
. . . . 5
⊢ (𝑣 = 𝐵 → ((𝐴 ∪ 𝑣) ∈ 𝑆 ↔ (𝐴 ∪ 𝐵) ∈ 𝑆)) |
14 | | difeq2 4052 |
. . . . . 6
⊢ (𝑣 = 𝐵 → (𝐴 ∖ 𝑣) = (𝐴 ∖ 𝐵)) |
15 | 14 | eleq1d 2823 |
. . . . 5
⊢ (𝑣 = 𝐵 → ((𝐴 ∖ 𝑣) ∈ 𝑆 ↔ (𝐴 ∖ 𝐵) ∈ 𝑆)) |
16 | 13, 15 | anbi12d 631 |
. . . 4
⊢ (𝑣 = 𝐵 → (((𝐴 ∪ 𝑣) ∈ 𝑆 ∧ (𝐴 ∖ 𝑣) ∈ 𝑆) ↔ ((𝐴 ∪ 𝐵) ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆))) |
17 | 11, 16 | rspc2va 3572 |
. . 3
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)) → ((𝐴 ∪ 𝐵) ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆)) |
18 | 1, 2, 6, 17 | syl21anc 835 |
. 2
⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ∪ 𝐵) ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆)) |
19 | 18 | simpld 495 |
1
⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆) |