Step |
Hyp |
Ref |
Expression |
1 |
|
chrelat2 |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐴 ⊆ 𝐵 ↔ ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) ) |
2 |
|
dfrex2 |
⊢ ( ∃ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ↔ ¬ ∀ 𝑥 ∈ HAtoms ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
3 |
1 2
|
bitrdi |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐴 ⊆ 𝐵 ↔ ¬ ∀ 𝑥 ∈ HAtoms ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) ) |
4 |
3
|
con4bid |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) ) |
5 |
|
iman |
⊢ ( ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ↔ ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
6 |
5
|
ralbii |
⊢ ( ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ↔ ∀ 𝑥 ∈ HAtoms ¬ ( 𝑥 ⊆ 𝐴 ∧ ¬ 𝑥 ⊆ 𝐵 ) ) |
7 |
4 6
|
bitr4di |
⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ HAtoms ( 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵 ) ) ) |