Step |
Hyp |
Ref |
Expression |
1 |
|
ssltex1 |
⊢ ( 𝐴 <<s 𝐵 → 𝐴 ∈ V ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐴 ∈ V ) |
3 |
|
ssltex2 |
⊢ ( 𝐴 <<s 𝐵 → 𝐵 ∈ V ) |
4 |
3
|
adantr |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐵 ∈ V ) |
5 |
|
simpr |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 ⊆ 𝐵 ) |
6 |
4 5
|
ssexd |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 ∈ V ) |
7 |
|
ssltss1 |
⊢ ( 𝐴 <<s 𝐵 → 𝐴 ⊆ No ) |
8 |
7
|
adantr |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐴 ⊆ No ) |
9 |
|
ssltss2 |
⊢ ( 𝐴 <<s 𝐵 → 𝐵 ⊆ No ) |
10 |
9
|
adantr |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐵 ⊆ No ) |
11 |
5 10
|
sstrd |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 ⊆ No ) |
12 |
|
ssltsep |
⊢ ( 𝐴 <<s 𝐵 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 ) |
13 |
|
ssralv |
⊢ ( 𝐶 ⊆ 𝐵 → ( ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀ 𝑦 ∈ 𝐶 𝑥 <s 𝑦 ) ) |
14 |
13
|
ralimdv |
⊢ ( 𝐶 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝑥 <s 𝑦 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝑥 <s 𝑦 ) ) |
15 |
12 14
|
mpan9 |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝑥 <s 𝑦 ) |
16 |
8 11 15
|
3jca |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → ( 𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝑥 <s 𝑦 ) ) |
17 |
|
brsslt |
⊢ ( 𝐴 <<s 𝐶 ↔ ( ( 𝐴 ∈ V ∧ 𝐶 ∈ V ) ∧ ( 𝐴 ⊆ No ∧ 𝐶 ⊆ No ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐶 𝑥 <s 𝑦 ) ) ) |
18 |
2 6 16 17
|
syl21anbrc |
⊢ ( ( 𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → 𝐴 <<s 𝐶 ) |