Step |
Hyp |
Ref |
Expression |
1 |
|
madef |
⊢ M : On ⟶ 𝒫 No |
2 |
1
|
ffvelrni |
⊢ ( 𝐴 ∈ On → ( M ‘ 𝐴 ) ∈ 𝒫 No ) |
3 |
2
|
elpwid |
⊢ ( 𝐴 ∈ On → ( M ‘ 𝐴 ) ⊆ No ) |
4 |
3
|
sseld |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) → 𝑋 ∈ No ) ) |
5 |
|
scutcl |
⊢ ( 𝑙 <<s 𝑟 → ( 𝑙 |s 𝑟 ) ∈ No ) |
6 |
|
eleq1 |
⊢ ( ( 𝑙 |s 𝑟 ) = 𝑋 → ( ( 𝑙 |s 𝑟 ) ∈ No ↔ 𝑋 ∈ No ) ) |
7 |
6
|
biimpd |
⊢ ( ( 𝑙 |s 𝑟 ) = 𝑋 → ( ( 𝑙 |s 𝑟 ) ∈ No → 𝑋 ∈ No ) ) |
8 |
5 7
|
mpan9 |
⊢ ( ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 ∈ No ) |
9 |
8
|
rexlimivw |
⊢ ( ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 ∈ No ) |
10 |
9
|
rexlimivw |
⊢ ( ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 ∈ No ) |
11 |
10
|
a1i |
⊢ ( 𝐴 ∈ On → ( ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) → 𝑋 ∈ No ) ) |
12 |
|
madeval2 |
⊢ ( 𝐴 ∈ On → ( M ‘ 𝐴 ) = { 𝑥 ∈ No ∣ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) } ) |
13 |
12
|
eleq2d |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ 𝑋 ∈ { 𝑥 ∈ No ∣ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) } ) ) |
14 |
|
eqeq2 |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑙 |s 𝑟 ) = 𝑥 ↔ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) |
15 |
14
|
anbi2d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) ↔ ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) ) |
16 |
15
|
2rexbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) ↔ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) ) |
17 |
16
|
elrab3 |
⊢ ( 𝑋 ∈ No → ( 𝑋 ∈ { 𝑥 ∈ No ∣ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑥 ) } ↔ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) ) |
18 |
13 17
|
sylan9bb |
⊢ ( ( 𝐴 ∈ On ∧ 𝑋 ∈ No ) → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) ) |
19 |
18
|
ex |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ No → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) ) ) |
20 |
4 11 19
|
pm5.21ndd |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( M ‘ 𝐴 ) ↔ ∃ 𝑙 ∈ 𝒫 ∪ ( M “ 𝐴 ) ∃ 𝑟 ∈ 𝒫 ∪ ( M “ 𝐴 ) ( 𝑙 <<s 𝑟 ∧ ( 𝑙 |s 𝑟 ) = 𝑋 ) ) ) |