Step |
Hyp |
Ref |
Expression |
1 |
|
sltlen.1 |
⊢ ( 𝜑 → 𝐴 ∈ No ) |
2 |
|
sltlen.2 |
⊢ ( 𝜑 → 𝐵 ∈ No ) |
3 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 <s 𝐵 ) → 𝐴 ∈ No ) |
4 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 <s 𝐵 ) → 𝐵 ∈ No ) |
5 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 <s 𝐵 ) → 𝐴 <s 𝐵 ) |
6 |
3 4 5
|
sltled |
⊢ ( ( 𝜑 ∧ 𝐴 <s 𝐵 ) → 𝐴 ≤s 𝐵 ) |
7 |
6
|
ex |
⊢ ( 𝜑 → ( 𝐴 <s 𝐵 → 𝐴 ≤s 𝐵 ) ) |
8 |
|
sltne |
⊢ ( ( 𝐴 ∈ No ∧ 𝐴 <s 𝐵 ) → 𝐵 ≠ 𝐴 ) |
9 |
1 8
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐴 <s 𝐵 ) → 𝐵 ≠ 𝐴 ) |
10 |
9
|
ex |
⊢ ( 𝜑 → ( 𝐴 <s 𝐵 → 𝐵 ≠ 𝐴 ) ) |
11 |
7 10
|
jcad |
⊢ ( 𝜑 → ( 𝐴 <s 𝐵 → ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) ) |
12 |
|
sleloe |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ) ) ) |
13 |
1 2 12
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 ↔ ( 𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ) ) ) |
14 |
|
eqneqall |
⊢ ( 𝐵 = 𝐴 → ( 𝐵 ≠ 𝐴 → 𝐴 <s 𝐵 ) ) |
15 |
14
|
eqcoms |
⊢ ( 𝐴 = 𝐵 → ( 𝐵 ≠ 𝐴 → 𝐴 <s 𝐵 ) ) |
16 |
15
|
jao1i |
⊢ ( ( 𝐴 <s 𝐵 ∨ 𝐴 = 𝐵 ) → ( 𝐵 ≠ 𝐴 → 𝐴 <s 𝐵 ) ) |
17 |
13 16
|
biimtrdi |
⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 → ( 𝐵 ≠ 𝐴 → 𝐴 <s 𝐵 ) ) ) |
18 |
17
|
impd |
⊢ ( 𝜑 → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴 ) → 𝐴 <s 𝐵 ) ) |
19 |
11 18
|
impbid |
⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≠ 𝐴 ) ) ) |