Step |
Hyp |
Ref |
Expression |
1 |
|
cutmin.1 |
⊢ ( 𝜑 → 𝐴 <<s 𝐵 ) |
2 |
|
cutmin.2 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
3 |
|
cutmin.3 |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 𝑋 ≤s 𝑦 ) |
4 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
5 |
|
ssltss1 |
⊢ ( 𝐴 <<s 𝐵 → 𝐴 ⊆ No ) |
6 |
1 5
|
syl |
⊢ ( 𝜑 → 𝐴 ⊆ No ) |
7 |
6
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ No ) |
8 |
|
slerflex |
⊢ ( 𝑥 ∈ No → 𝑥 ≤s 𝑥 ) |
9 |
7 8
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ≤s 𝑥 ) |
10 |
|
breq2 |
⊢ ( 𝑦 = 𝑥 → ( 𝑥 ≤s 𝑦 ↔ 𝑥 ≤s 𝑥 ) ) |
11 |
10
|
rspcev |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≤s 𝑥 ) → ∃ 𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ) |
12 |
4 9 11
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ) |
13 |
12
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ) |
14 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤s 𝑦 ↔ 𝑋 ≤s 𝑦 ) ) |
15 |
14
|
rexsng |
⊢ ( 𝑋 ∈ 𝐵 → ( ∃ 𝑥 ∈ { 𝑋 } 𝑥 ≤s 𝑦 ↔ 𝑋 ≤s 𝑦 ) ) |
16 |
2 15
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ { 𝑋 } 𝑥 ≤s 𝑦 ↔ 𝑋 ≤s 𝑦 ) ) |
17 |
16
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ { 𝑋 } 𝑥 ≤s 𝑦 ↔ ∀ 𝑦 ∈ 𝐵 𝑋 ≤s 𝑦 ) ) |
18 |
3 17
|
mpbird |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ { 𝑋 } 𝑥 ≤s 𝑦 ) |
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 |
20
|
simp3d |
⊢ ( 𝜑 → { ( 𝐴 |s 𝐵 ) } <<s 𝐵 ) |
23 |
2
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ 𝐵 ) |
24 |
|
sssslt2 |
⊢ ( ( { ( 𝐴 |s 𝐵 ) } <<s 𝐵 ∧ { 𝑋 } ⊆ 𝐵 ) → { ( 𝐴 |s 𝐵 ) } <<s { 𝑋 } ) |
25 |
22 23 24
|
syl2anc |
⊢ ( 𝜑 → { ( 𝐴 |s 𝐵 ) } <<s { 𝑋 } ) |
26 |
1 13 18 21 25
|
cofcut1d |
⊢ ( 𝜑 → ( 𝐴 |s 𝐵 ) = ( 𝐴 |s { 𝑋 } ) ) |