Step | Hyp | Ref
| Expression |
1 | | elex 3426 |
. 2
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ V) |
2 | | elex 3426 |
. . 3
⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
3 | 2 | adantr 484 |
. 2
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ V) |
4 | | eleq1 2825 |
. . . 4
⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
5 | | eleq1 2825 |
. . . . 5
⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) |
6 | 5 | notbid 321 |
. . . 4
⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)) |
7 | 4, 6 | anbi12d 634 |
. . 3
⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))) |
8 | | df-dif 3869 |
. . 3
⊢ (𝐵 ∖ 𝐶) = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)} |
9 | 7, 8 | elab2g 3589 |
. 2
⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))) |
10 | 1, 3, 9 | pm5.21nii 383 |
1
⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |