Step |
Hyp |
Ref |
Expression |
1 |
|
xpnz |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅ ) ↔ ( 𝐴 × 𝐵 ) ≠ ∅ ) |
2 |
|
anidm |
⊢ ( ( ( 𝐴 × 𝐵 ) ≠ ∅ ∧ ( 𝐴 × 𝐵 ) ≠ ∅ ) ↔ ( 𝐴 × 𝐵 ) ≠ ∅ ) |
3 |
|
neeq1 |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( 𝐴 × 𝐵 ) ≠ ∅ ↔ ( 𝐶 × 𝐷 ) ≠ ∅ ) ) |
4 |
3
|
anbi2d |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( ( 𝐴 × 𝐵 ) ≠ ∅ ∧ ( 𝐴 × 𝐵 ) ≠ ∅ ) ↔ ( ( 𝐴 × 𝐵 ) ≠ ∅ ∧ ( 𝐶 × 𝐷 ) ≠ ∅ ) ) ) |
5 |
2 4
|
bitr3id |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( 𝐴 × 𝐵 ) ≠ ∅ ↔ ( ( 𝐴 × 𝐵 ) ≠ ∅ ∧ ( 𝐶 × 𝐷 ) ≠ ∅ ) ) ) |
6 |
|
eqimss |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( 𝐴 × 𝐵 ) ⊆ ( 𝐶 × 𝐷 ) ) |
7 |
|
ssxpb |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ( ( 𝐴 × 𝐵 ) ⊆ ( 𝐶 × 𝐷 ) ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷 ) ) ) |
8 |
6 7
|
syl5ibcom |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( 𝐴 × 𝐵 ) ≠ ∅ → ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷 ) ) ) |
9 |
|
eqimss2 |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( 𝐶 × 𝐷 ) ⊆ ( 𝐴 × 𝐵 ) ) |
10 |
|
ssxpb |
⊢ ( ( 𝐶 × 𝐷 ) ≠ ∅ → ( ( 𝐶 × 𝐷 ) ⊆ ( 𝐴 × 𝐵 ) ↔ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) ) ) |
11 |
9 10
|
syl5ibcom |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( 𝐶 × 𝐷 ) ≠ ∅ → ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) ) ) |
12 |
8 11
|
anim12d |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( ( 𝐴 × 𝐵 ) ≠ ∅ ∧ ( 𝐶 × 𝐷 ) ≠ ∅ ) → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷 ) ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) ) ) ) |
13 |
|
an4 |
⊢ ( ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷 ) ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) ) ↔ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( 𝐵 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) ) |
14 |
|
eqss |
⊢ ( 𝐴 = 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ) |
15 |
|
eqss |
⊢ ( 𝐵 = 𝐷 ↔ ( 𝐵 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) |
16 |
14 15
|
anbi12i |
⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ↔ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴 ) ∧ ( 𝐵 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵 ) ) ) |
17 |
13 16
|
bitr4i |
⊢ ( ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷 ) ∧ ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) |
18 |
12 17
|
syl6ib |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( ( 𝐴 × 𝐵 ) ≠ ∅ ∧ ( 𝐶 × 𝐷 ) ≠ ∅ ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
19 |
5 18
|
sylbid |
⊢ ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( ( 𝐴 × 𝐵 ) ≠ ∅ → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
20 |
19
|
com12 |
⊢ ( ( 𝐴 × 𝐵 ) ≠ ∅ → ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
21 |
1 20
|
sylbi |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅ ) → ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) → ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |
22 |
|
xpeq12 |
⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) ) |
23 |
21 22
|
impbid1 |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅ ) → ( ( 𝐴 × 𝐵 ) = ( 𝐶 × 𝐷 ) ↔ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ) |