Step |
Hyp |
Ref |
Expression |
1 |
|
enrefg |
⊢ ( 𝐴 ∈ On → 𝐴 ≈ 𝐴 ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐴 ≈ 𝐴 ) |
3 |
|
simpr |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐵 ∈ On ) |
4 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) |
5 |
4
|
oacomf1olem |
⊢ ( ( 𝐵 ∈ On ∧ 𝐴 ∈ On ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) ) |
6 |
5
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) ) |
7 |
6
|
simpld |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) |
8 |
|
f1oeng |
⊢ ( ( 𝐵 ∈ On ∧ ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) : 𝐵 –1-1-onto→ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) → 𝐵 ≈ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) |
9 |
3 7 8
|
syl2anc |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → 𝐵 ≈ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) |
10 |
|
incom |
⊢ ( 𝐴 ∩ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) |
11 |
6
|
simprd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∩ 𝐴 ) = ∅ ) |
12 |
10 11
|
eqtrid |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ∩ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ∅ ) |
13 |
|
djuenun |
⊢ ( ( 𝐴 ≈ 𝐴 ∧ 𝐵 ≈ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ∧ ( 𝐴 ∩ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) = ∅ ) → ( 𝐴 ⊔ 𝐵 ) ≈ ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) ) |
14 |
2 9 12 13
|
syl3anc |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊔ 𝐵 ) ≈ ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) ) |
15 |
|
oarec |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) = ( 𝐴 ∪ ran ( 𝑥 ∈ 𝐵 ↦ ( 𝐴 +o 𝑥 ) ) ) ) |
16 |
14 15
|
breqtrrd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊔ 𝐵 ) ≈ ( 𝐴 +o 𝐵 ) ) |
17 |
16
|
ensymd |
⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 +o 𝐵 ) ≈ ( 𝐴 ⊔ 𝐵 ) ) |