Proof of Theorem oneqmini
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | ssint 4963 | . . . . . 6
⊢ (𝐴 ⊆ ∩ 𝐵
↔ ∀𝑥 ∈
𝐵 𝐴 ⊆ 𝑥) | 
| 2 |  | ssel 3976 | . . . . . . . . . . . 12
⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On)) | 
| 3 |  | ssel 3976 | . . . . . . . . . . . 12
⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On)) | 
| 4 | 2, 3 | anim12d 609 | . . . . . . . . . . 11
⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On))) | 
| 5 |  | ontri1 6417 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴)) | 
| 6 | 4, 5 | syl6 35 | . . . . . . . . . 10
⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))) | 
| 7 | 6 | expdimp 452 | . . . . . . . . 9
⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ 𝐵 → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))) | 
| 8 | 7 | pm5.74d 273 | . . . . . . . 8
⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))) | 
| 9 |  | con2b 359 | . . . . . . . 8
⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) | 
| 10 | 8, 9 | bitrdi 287 | . . . . . . 7
⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))) | 
| 11 | 10 | ralbidv2 3173 | . . . . . 6
⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)) | 
| 12 | 1, 11 | bitrid 283 | . . . . 5
⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)) | 
| 13 | 12 | biimprd 248 | . . . 4
⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∩ 𝐵)) | 
| 14 | 13 | expimpd 453 | . . 3
⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ∩ 𝐵)) | 
| 15 |  | intss1 4962 | . . . . 5
⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴) | 
| 16 | 15 | a1i 11 | . . . 4
⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)) | 
| 17 | 16 | adantrd 491 | . . 3
⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → ∩ 𝐵 ⊆ 𝐴)) | 
| 18 | 14, 17 | jcad 512 | . 2
⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵
⊆ 𝐴))) | 
| 19 |  | eqss 3998 | . 2
⊢ (𝐴 = ∩
𝐵 ↔ (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵
⊆ 𝐴)) | 
| 20 | 18, 19 | imbitrrdi 252 | 1
⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵)) |