Step |
Hyp |
Ref |
Expression |
1 |
|
ssequn1 |
⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐵 ) |
2 |
|
eleq1a |
⊢ ( 𝐵 ∈ ω → ( ( 𝐴 ∪ 𝐵 ) = 𝐵 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) ) |
3 |
2
|
adantl |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ∪ 𝐵 ) = 𝐵 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) ) |
4 |
1 3
|
biimtrid |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) ) |
5 |
|
ssequn2 |
⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐴 ) |
6 |
|
eleq1a |
⊢ ( 𝐴 ∈ ω → ( ( 𝐴 ∪ 𝐵 ) = 𝐴 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝐴 ∪ 𝐵 ) = 𝐴 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) ) |
8 |
5 7
|
biimtrid |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∪ 𝐵 ) ∈ ω ) ) |
9 |
|
nnord |
⊢ ( 𝐴 ∈ ω → Ord 𝐴 ) |
10 |
|
nnord |
⊢ ( 𝐵 ∈ ω → Ord 𝐵 ) |
11 |
|
ordtri2or2 |
⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ) |
12 |
9 10 11
|
syl2an |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ) |
13 |
4 8 12
|
mpjaod |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐴 ∪ 𝐵 ) ∈ ω ) |