Step |
Hyp |
Ref |
Expression |
1 |
|
df-dju |
⊢ ( 𝐴 ⊔ 𝐵 ) = ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐵 ) ) |
2 |
|
0ex |
⊢ ∅ ∈ V |
3 |
|
relsdom |
⊢ Rel ≺ |
4 |
3
|
brrelex2i |
⊢ ( 1o ≺ 𝐴 → 𝐴 ∈ V ) |
5 |
|
xpsnen2g |
⊢ ( ( ∅ ∈ V ∧ 𝐴 ∈ V ) → ( { ∅ } × 𝐴 ) ≈ 𝐴 ) |
6 |
2 4 5
|
sylancr |
⊢ ( 1o ≺ 𝐴 → ( { ∅ } × 𝐴 ) ≈ 𝐴 ) |
7 |
|
sdomen2 |
⊢ ( ( { ∅ } × 𝐴 ) ≈ 𝐴 → ( 1o ≺ ( { ∅ } × 𝐴 ) ↔ 1o ≺ 𝐴 ) ) |
8 |
6 7
|
syl |
⊢ ( 1o ≺ 𝐴 → ( 1o ≺ ( { ∅ } × 𝐴 ) ↔ 1o ≺ 𝐴 ) ) |
9 |
8
|
ibir |
⊢ ( 1o ≺ 𝐴 → 1o ≺ ( { ∅ } × 𝐴 ) ) |
10 |
|
1on |
⊢ 1o ∈ On |
11 |
3
|
brrelex2i |
⊢ ( 1o ≺ 𝐵 → 𝐵 ∈ V ) |
12 |
|
xpsnen2g |
⊢ ( ( 1o ∈ On ∧ 𝐵 ∈ V ) → ( { 1o } × 𝐵 ) ≈ 𝐵 ) |
13 |
10 11 12
|
sylancr |
⊢ ( 1o ≺ 𝐵 → ( { 1o } × 𝐵 ) ≈ 𝐵 ) |
14 |
|
sdomen2 |
⊢ ( ( { 1o } × 𝐵 ) ≈ 𝐵 → ( 1o ≺ ( { 1o } × 𝐵 ) ↔ 1o ≺ 𝐵 ) ) |
15 |
13 14
|
syl |
⊢ ( 1o ≺ 𝐵 → ( 1o ≺ ( { 1o } × 𝐵 ) ↔ 1o ≺ 𝐵 ) ) |
16 |
15
|
ibir |
⊢ ( 1o ≺ 𝐵 → 1o ≺ ( { 1o } × 𝐵 ) ) |
17 |
|
unxpdom |
⊢ ( ( 1o ≺ ( { ∅ } × 𝐴 ) ∧ 1o ≺ ( { 1o } × 𝐵 ) ) → ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐵 ) ) ≼ ( ( { ∅ } × 𝐴 ) × ( { 1o } × 𝐵 ) ) ) |
18 |
9 16 17
|
syl2an |
⊢ ( ( 1o ≺ 𝐴 ∧ 1o ≺ 𝐵 ) → ( ( { ∅ } × 𝐴 ) ∪ ( { 1o } × 𝐵 ) ) ≼ ( ( { ∅ } × 𝐴 ) × ( { 1o } × 𝐵 ) ) ) |
19 |
1 18
|
eqbrtrid |
⊢ ( ( 1o ≺ 𝐴 ∧ 1o ≺ 𝐵 ) → ( 𝐴 ⊔ 𝐵 ) ≼ ( ( { ∅ } × 𝐴 ) × ( { 1o } × 𝐵 ) ) ) |
20 |
|
xpen |
⊢ ( ( ( { ∅ } × 𝐴 ) ≈ 𝐴 ∧ ( { 1o } × 𝐵 ) ≈ 𝐵 ) → ( ( { ∅ } × 𝐴 ) × ( { 1o } × 𝐵 ) ) ≈ ( 𝐴 × 𝐵 ) ) |
21 |
6 13 20
|
syl2an |
⊢ ( ( 1o ≺ 𝐴 ∧ 1o ≺ 𝐵 ) → ( ( { ∅ } × 𝐴 ) × ( { 1o } × 𝐵 ) ) ≈ ( 𝐴 × 𝐵 ) ) |
22 |
|
domentr |
⊢ ( ( ( 𝐴 ⊔ 𝐵 ) ≼ ( ( { ∅ } × 𝐴 ) × ( { 1o } × 𝐵 ) ) ∧ ( ( { ∅ } × 𝐴 ) × ( { 1o } × 𝐵 ) ) ≈ ( 𝐴 × 𝐵 ) ) → ( 𝐴 ⊔ 𝐵 ) ≼ ( 𝐴 × 𝐵 ) ) |
23 |
19 21 22
|
syl2anc |
⊢ ( ( 1o ≺ 𝐴 ∧ 1o ≺ 𝐵 ) → ( 𝐴 ⊔ 𝐵 ) ≼ ( 𝐴 × 𝐵 ) ) |