Step |
Hyp |
Ref |
Expression |
1 |
|
cutmax.1 |
⊢ ( 𝜑 → 𝐴 <<s 𝐵 ) |
2 |
|
cutmax.2 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐴 ) |
3 |
|
cutmax.3 |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 𝑦 ≤s 𝑋 ) |
4 |
|
breq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑋 ) ) |
5 |
4
|
rexsng |
⊢ ( 𝑋 ∈ 𝐴 → ( ∃ 𝑥 ∈ { 𝑋 } 𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑋 ) ) |
6 |
2 5
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ { 𝑋 } 𝑦 ≤s 𝑥 ↔ 𝑦 ≤s 𝑋 ) ) |
7 |
6
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ { 𝑋 } 𝑦 ≤s 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ≤s 𝑋 ) ) |
8 |
3 7
|
mpbird |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐴 ∃ 𝑥 ∈ { 𝑋 } 𝑦 ≤s 𝑥 ) |
9 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) |
10 |
|
ssltss2 |
⊢ ( 𝐴 <<s 𝐵 → 𝐵 ⊆ No ) |
11 |
1 10
|
syl |
⊢ ( 𝜑 → 𝐵 ⊆ No ) |
12 |
11
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ No ) |
13 |
|
slerflex |
⊢ ( 𝑥 ∈ No → 𝑥 ≤s 𝑥 ) |
14 |
12 13
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ≤s 𝑥 ) |
15 |
|
breq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ≤s 𝑥 ↔ 𝑥 ≤s 𝑥 ) ) |
16 |
15
|
rspcev |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ≤s 𝑥 ) → ∃ 𝑦 ∈ 𝐵 𝑦 ≤s 𝑥 ) |
17 |
9 14 16
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑦 ∈ 𝐵 𝑦 ≤s 𝑥 ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝑦 ≤s 𝑥 ) |
19 |
|
scutcut |
⊢ ( 𝐴 <<s 𝐵 → ( ( 𝐴 |s 𝐵 ) ∈ No ∧ 𝐴 <<s { ( 𝐴 |s 𝐵 ) } ∧ { ( 𝐴 |s 𝐵 ) } <<s 𝐵 ) ) |
20 |
1 19
|
syl |
⊢ ( 𝜑 → ( ( 𝐴 |s 𝐵 ) ∈ No ∧ 𝐴 <<s { ( 𝐴 |s 𝐵 ) } ∧ { ( 𝐴 |s 𝐵 ) } <<s 𝐵 ) ) |
21 |
20
|
simp2d |
⊢ ( 𝜑 → 𝐴 <<s { ( 𝐴 |s 𝐵 ) } ) |
22 |
2
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝐴 ) |
23 |
|
sssslt1 |
⊢ ( ( 𝐴 <<s { ( 𝐴 |s 𝐵 ) } ∧ { 𝑋 } ⊆ 𝐴 ) → { 𝑋 } <<s { ( 𝐴 |s 𝐵 ) } ) |
24 |
21 22 23
|
syl2anc |
⊢ ( 𝜑 → { 𝑋 } <<s { ( 𝐴 |s 𝐵 ) } ) |
25 |
20
|
simp3d |
⊢ ( 𝜑 → { ( 𝐴 |s 𝐵 ) } <<s 𝐵 ) |
26 |
1 8 18 24 25
|
cofcut1d |
⊢ ( 𝜑 → ( 𝐴 |s 𝐵 ) = ( { 𝑋 } |s 𝐵 ) ) |