Step |
Hyp |
Ref |
Expression |
1 |
|
hsmexlem4.x |
⊢ 𝑋 ∈ V |
2 |
|
hsmexlem4.h |
⊢ 𝐻 = ( rec ( ( 𝑧 ∈ V ↦ ( har ‘ 𝒫 ( 𝑋 × 𝑧 ) ) ) , ( har ‘ 𝒫 𝑋 ) ) ↾ ω ) |
3 |
|
hsmexlem4.u |
⊢ 𝑈 = ( 𝑥 ∈ V ↦ ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝑥 ) ↾ ω ) ) |
4 |
|
hsmexlem4.s |
⊢ 𝑆 = { 𝑎 ∈ ∪ ( 𝑅1 “ On ) ∣ ∀ 𝑏 ∈ ( TC ‘ { 𝑎 } ) 𝑏 ≼ 𝑋 } |
5 |
|
hsmexlem4.o |
⊢ 𝑂 = OrdIso ( E , ( rank “ ( ( 𝑈 ‘ 𝑑 ) ‘ 𝑐 ) ) ) |
6 |
|
fvex |
⊢ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) ∈ V |
7 |
1 2 3 4 5
|
hsmexlem5 |
⊢ ( 𝑑 ∈ 𝑆 → ( rank ‘ 𝑑 ) ∈ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) |
8 |
4
|
ssrab3 |
⊢ 𝑆 ⊆ ∪ ( 𝑅1 “ On ) |
9 |
8
|
sseli |
⊢ ( 𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ ( 𝑅1 “ On ) ) |
10 |
|
harcl |
⊢ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ∈ On |
11 |
|
r1fnon |
⊢ 𝑅1 Fn On |
12 |
11
|
fndmi |
⊢ dom 𝑅1 = On |
13 |
10 12
|
eleqtrri |
⊢ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ∈ dom 𝑅1 |
14 |
|
rankr1ag |
⊢ ( ( 𝑑 ∈ ∪ ( 𝑅1 “ On ) ∧ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ∈ dom 𝑅1 ) → ( 𝑑 ∈ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) ↔ ( rank ‘ 𝑑 ) ∈ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) ) |
15 |
9 13 14
|
sylancl |
⊢ ( 𝑑 ∈ 𝑆 → ( 𝑑 ∈ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) ↔ ( rank ‘ 𝑑 ) ∈ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) ) |
16 |
7 15
|
mpbird |
⊢ ( 𝑑 ∈ 𝑆 → 𝑑 ∈ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) ) |
17 |
16
|
ssriv |
⊢ 𝑆 ⊆ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ∪ ran 𝐻 ) ) ) |
18 |
6 17
|
ssexi |
⊢ 𝑆 ∈ V |