Step |
Hyp |
Ref |
Expression |
1 |
|
simp3 |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → 𝐶 ∈ On ) |
2 |
|
0elon |
⊢ ∅ ∈ On |
3 |
|
ordelon |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ On ) |
4 |
3
|
3ad2antl1 |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ On ) |
5 |
|
naddcom |
⊢ ( ( ∅ ∈ On ∧ 𝑥 ∈ On ) → ( ∅ +no 𝑥 ) = ( 𝑥 +no ∅ ) ) |
6 |
2 4 5
|
sylancr |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → ( ∅ +no 𝑥 ) = ( 𝑥 +no ∅ ) ) |
7 |
|
naddrid |
⊢ ( 𝑥 ∈ On → ( 𝑥 +no ∅ ) = 𝑥 ) |
8 |
4 7
|
syl |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 +no ∅ ) = 𝑥 ) |
9 |
6 8
|
eqtrd |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → ( ∅ +no 𝑥 ) = 𝑥 ) |
10 |
|
0ss |
⊢ ∅ ⊆ 𝐵 |
11 |
|
simpl2 |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ On ) |
12 |
|
naddssim |
⊢ ( ( ∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝑥 ∈ On ) → ( ∅ ⊆ 𝐵 → ( ∅ +no 𝑥 ) ⊆ ( 𝐵 +no 𝑥 ) ) ) |
13 |
2 11 4 12
|
mp3an2i |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → ( ∅ ⊆ 𝐵 → ( ∅ +no 𝑥 ) ⊆ ( 𝐵 +no 𝑥 ) ) ) |
14 |
10 13
|
mpi |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → ( ∅ +no 𝑥 ) ⊆ ( 𝐵 +no 𝑥 ) ) |
15 |
9 14
|
eqsstrrd |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ⊆ ( 𝐵 +no 𝑥 ) ) |
16 |
|
simpl3 |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ On ) |
17 |
|
ontr2 |
⊢ ( ( 𝑥 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝑥 ⊆ ( 𝐵 +no 𝑥 ) ∧ ( 𝐵 +no 𝑥 ) ∈ 𝐶 ) → 𝑥 ∈ 𝐶 ) ) |
18 |
4 16 17
|
syl2anc |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ⊆ ( 𝐵 +no 𝑥 ) ∧ ( 𝐵 +no 𝑥 ) ∈ 𝐶 ) → 𝑥 ∈ 𝐶 ) ) |
19 |
15 18
|
mpand |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 +no 𝑥 ) ∈ 𝐶 → 𝑥 ∈ 𝐶 ) ) |
20 |
19
|
3impia |
⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝑥 ∈ 𝐴 ∧ ( 𝐵 +no 𝑥 ) ∈ 𝐶 ) → 𝑥 ∈ 𝐶 ) |
21 |
20
|
rabssdv |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → { 𝑥 ∈ 𝐴 ∣ ( 𝐵 +no 𝑥 ) ∈ 𝐶 } ⊆ 𝐶 ) |
22 |
1 21
|
ssexd |
⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → { 𝑥 ∈ 𝐴 ∣ ( 𝐵 +no 𝑥 ) ∈ 𝐶 } ∈ V ) |