Step |
Hyp |
Ref |
Expression |
1 |
|
reldom |
⊢ Rel ≼ |
2 |
1
|
brrelex2i |
⊢ ( ω ≼ 𝐴 → 𝐴 ∈ V ) |
3 |
|
domeng |
⊢ ( 𝐴 ∈ V → ( ω ≼ 𝐴 ↔ ∃ 𝑡 ( ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ) ) |
4 |
|
bren |
⊢ ( ω ≈ 𝑡 ↔ ∃ ℎ ℎ : ω –1-1-onto→ 𝑡 ) |
5 |
|
harcl |
⊢ ( har ‘ 𝒫 𝐴 ) ∈ On |
6 |
|
infxpenc2 |
⊢ ( ( har ‘ 𝒫 𝐴 ) ∈ On → ∃ 𝑚 ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) |
7 |
5 6
|
ax-mp |
⊢ ∃ 𝑚 ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) |
8 |
|
simpr |
⊢ ( ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) ∧ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) → 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
9 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐴 ↑m 𝑛 ) = ( 𝐴 ↑m 𝑘 ) ) |
10 |
9
|
cbviunv |
⊢ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) = ∪ 𝑘 ∈ ω ( 𝐴 ↑m 𝑘 ) |
11 |
|
f1eq3 |
⊢ ( ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) = ∪ 𝑘 ∈ ω ( 𝐴 ↑m 𝑘 ) → ( 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↔ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑘 ∈ ω ( 𝐴 ↑m 𝑘 ) ) ) |
12 |
10 11
|
ax-mp |
⊢ ( 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ↔ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑘 ∈ ω ( 𝐴 ↑m 𝑘 ) ) |
13 |
8 12
|
sylib |
⊢ ( ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) ∧ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) → 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑘 ∈ ω ( 𝐴 ↑m 𝑘 ) ) |
14 |
|
simpllr |
⊢ ( ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) ∧ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) → 𝑡 ⊆ 𝐴 ) |
15 |
|
simplll |
⊢ ( ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) ∧ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) → ℎ : ω –1-1-onto→ 𝑡 ) |
16 |
|
biid |
⊢ ( ( ( 𝑢 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑢 × 𝑢 ) ∧ 𝑟 We 𝑢 ) ∧ ω ≼ 𝑢 ) ↔ ( ( 𝑢 ⊆ 𝐴 ∧ 𝑟 ⊆ ( 𝑢 × 𝑢 ) ∧ 𝑟 We 𝑢 ) ∧ ω ≼ 𝑢 ) ) |
17 |
|
simplr |
⊢ ( ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) ∧ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) → ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) |
18 |
|
sseq2 |
⊢ ( 𝑏 = 𝑤 → ( ω ⊆ 𝑏 ↔ ω ⊆ 𝑤 ) ) |
19 |
|
fveq2 |
⊢ ( 𝑏 = 𝑤 → ( 𝑚 ‘ 𝑏 ) = ( 𝑚 ‘ 𝑤 ) ) |
20 |
19
|
f1oeq1d |
⊢ ( 𝑏 = 𝑤 → ( ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ↔ ( 𝑚 ‘ 𝑤 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) |
21 |
|
xpeq12 |
⊢ ( ( 𝑏 = 𝑤 ∧ 𝑏 = 𝑤 ) → ( 𝑏 × 𝑏 ) = ( 𝑤 × 𝑤 ) ) |
22 |
21
|
anidms |
⊢ ( 𝑏 = 𝑤 → ( 𝑏 × 𝑏 ) = ( 𝑤 × 𝑤 ) ) |
23 |
22
|
f1oeq2d |
⊢ ( 𝑏 = 𝑤 → ( ( 𝑚 ‘ 𝑤 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ↔ ( 𝑚 ‘ 𝑤 ) : ( 𝑤 × 𝑤 ) –1-1-onto→ 𝑏 ) ) |
24 |
|
f1oeq3 |
⊢ ( 𝑏 = 𝑤 → ( ( 𝑚 ‘ 𝑤 ) : ( 𝑤 × 𝑤 ) –1-1-onto→ 𝑏 ↔ ( 𝑚 ‘ 𝑤 ) : ( 𝑤 × 𝑤 ) –1-1-onto→ 𝑤 ) ) |
25 |
20 23 24
|
3bitrd |
⊢ ( 𝑏 = 𝑤 → ( ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ↔ ( 𝑚 ‘ 𝑤 ) : ( 𝑤 × 𝑤 ) –1-1-onto→ 𝑤 ) ) |
26 |
18 25
|
imbi12d |
⊢ ( 𝑏 = 𝑤 → ( ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ↔ ( ω ⊆ 𝑤 → ( 𝑚 ‘ 𝑤 ) : ( 𝑤 × 𝑤 ) –1-1-onto→ 𝑤 ) ) ) |
27 |
26
|
cbvralvw |
⊢ ( ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ↔ ∀ 𝑤 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑤 → ( 𝑚 ‘ 𝑤 ) : ( 𝑤 × 𝑤 ) –1-1-onto→ 𝑤 ) ) |
28 |
17 27
|
sylib |
⊢ ( ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) ∧ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) → ∀ 𝑤 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑤 → ( 𝑚 ‘ 𝑤 ) : ( 𝑤 × 𝑤 ) –1-1-onto→ 𝑤 ) ) |
29 |
|
eqid |
⊢ OrdIso ( 𝑟 , 𝑢 ) = OrdIso ( 𝑟 , 𝑢 ) |
30 |
|
eqid |
⊢ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) = ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) |
31 |
|
eqid |
⊢ ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) = ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) |
32 |
|
eqid |
⊢ seqω ( ( 𝑝 ∈ V , 𝑓 ∈ V ↦ ( 𝑥 ∈ ( 𝑢 ↑m suc 𝑝 ) ↦ ( ( 𝑓 ‘ ( 𝑥 ↾ 𝑝 ) ) ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ( 𝑥 ‘ 𝑝 ) ) ) ) , { 〈 ∅ , ( OrdIso ( 𝑟 , 𝑢 ) ‘ ∅ ) 〉 } ) = seqω ( ( 𝑝 ∈ V , 𝑓 ∈ V ↦ ( 𝑥 ∈ ( 𝑢 ↑m suc 𝑝 ) ↦ ( ( 𝑓 ‘ ( 𝑥 ↾ 𝑝 ) ) ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ( 𝑥 ‘ 𝑝 ) ) ) ) , { 〈 ∅ , ( OrdIso ( 𝑟 , 𝑢 ) ‘ ∅ ) 〉 } ) |
33 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑢 ↑m 𝑛 ) = ( 𝑢 ↑m 𝑘 ) ) |
34 |
33
|
cbviunv |
⊢ ∪ 𝑛 ∈ ω ( 𝑢 ↑m 𝑛 ) = ∪ 𝑘 ∈ ω ( 𝑢 ↑m 𝑘 ) |
35 |
34
|
mpteq1i |
⊢ ( 𝑦 ∈ ∪ 𝑛 ∈ ω ( 𝑢 ↑m 𝑛 ) ↦ 〈 dom 𝑦 , ( ( seqω ( ( 𝑝 ∈ V , 𝑓 ∈ V ↦ ( 𝑥 ∈ ( 𝑢 ↑m suc 𝑝 ) ↦ ( ( 𝑓 ‘ ( 𝑥 ↾ 𝑝 ) ) ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ( 𝑥 ‘ 𝑝 ) ) ) ) , { 〈 ∅ , ( OrdIso ( 𝑟 , 𝑢 ) ‘ ∅ ) 〉 } ) ‘ dom 𝑦 ) ‘ 𝑦 ) 〉 ) = ( 𝑦 ∈ ∪ 𝑘 ∈ ω ( 𝑢 ↑m 𝑘 ) ↦ 〈 dom 𝑦 , ( ( seqω ( ( 𝑝 ∈ V , 𝑓 ∈ V ↦ ( 𝑥 ∈ ( 𝑢 ↑m suc 𝑝 ) ↦ ( ( 𝑓 ‘ ( 𝑥 ↾ 𝑝 ) ) ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ( 𝑥 ‘ 𝑝 ) ) ) ) , { 〈 ∅ , ( OrdIso ( 𝑟 , 𝑢 ) ‘ ∅ ) 〉 } ) ‘ dom 𝑦 ) ‘ 𝑦 ) 〉 ) |
36 |
|
eqid |
⊢ ( 𝑥 ∈ ω , 𝑦 ∈ 𝑢 ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑥 ) , 𝑦 〉 ) = ( 𝑥 ∈ ω , 𝑦 ∈ 𝑢 ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑥 ) , 𝑦 〉 ) |
37 |
|
eqid |
⊢ ( ( ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ∘ ( 𝑥 ∈ ω , 𝑦 ∈ 𝑢 ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑥 ) , 𝑦 〉 ) ) ∘ ( 𝑦 ∈ ∪ 𝑛 ∈ ω ( 𝑢 ↑m 𝑛 ) ↦ 〈 dom 𝑦 , ( ( seqω ( ( 𝑝 ∈ V , 𝑓 ∈ V ↦ ( 𝑥 ∈ ( 𝑢 ↑m suc 𝑝 ) ↦ ( ( 𝑓 ‘ ( 𝑥 ↾ 𝑝 ) ) ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ( 𝑥 ‘ 𝑝 ) ) ) ) , { 〈 ∅ , ( OrdIso ( 𝑟 , 𝑢 ) ‘ ∅ ) 〉 } ) ‘ dom 𝑦 ) ‘ 𝑦 ) 〉 ) ) = ( ( ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ∘ ( 𝑥 ∈ ω , 𝑦 ∈ 𝑢 ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑥 ) , 𝑦 〉 ) ) ∘ ( 𝑦 ∈ ∪ 𝑛 ∈ ω ( 𝑢 ↑m 𝑛 ) ↦ 〈 dom 𝑦 , ( ( seqω ( ( 𝑝 ∈ V , 𝑓 ∈ V ↦ ( 𝑥 ∈ ( 𝑢 ↑m suc 𝑝 ) ↦ ( ( 𝑓 ‘ ( 𝑥 ↾ 𝑝 ) ) ( ( OrdIso ( 𝑟 , 𝑢 ) ∘ ( 𝑚 ‘ dom OrdIso ( 𝑟 , 𝑢 ) ) ) ∘ ◡ ( 𝑠 ∈ dom OrdIso ( 𝑟 , 𝑢 ) , 𝑧 ∈ dom OrdIso ( 𝑟 , 𝑢 ) ↦ 〈 ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑠 ) , ( OrdIso ( 𝑟 , 𝑢 ) ‘ 𝑧 ) 〉 ) ) ( 𝑥 ‘ 𝑝 ) ) ) ) , { 〈 ∅ , ( OrdIso ( 𝑟 , 𝑢 ) ‘ ∅ ) 〉 } ) ‘ dom 𝑦 ) ‘ 𝑦 ) 〉 ) ) |
38 |
13 14 15 16 28 29 30 31 32 35 36 37
|
pwfseqlem5 |
⊢ ¬ ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) ∧ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
39 |
38
|
imnani |
⊢ ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) → ¬ 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
40 |
39
|
nexdv |
⊢ ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) → ¬ ∃ 𝑔 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
41 |
|
brdomi |
⊢ ( 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) → ∃ 𝑔 𝑔 : 𝒫 𝐴 –1-1→ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
42 |
40 41
|
nsyl |
⊢ ( ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) ∧ ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) ) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
43 |
42
|
ex |
⊢ ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) → ( ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) ) |
44 |
43
|
exlimdv |
⊢ ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) → ( ∃ 𝑚 ∀ 𝑏 ∈ ( har ‘ 𝒫 𝐴 ) ( ω ⊆ 𝑏 → ( 𝑚 ‘ 𝑏 ) : ( 𝑏 × 𝑏 ) –1-1-onto→ 𝑏 ) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) ) |
45 |
7 44
|
mpi |
⊢ ( ( ℎ : ω –1-1-onto→ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
46 |
45
|
ex |
⊢ ( ℎ : ω –1-1-onto→ 𝑡 → ( 𝑡 ⊆ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) ) |
47 |
46
|
exlimiv |
⊢ ( ∃ ℎ ℎ : ω –1-1-onto→ 𝑡 → ( 𝑡 ⊆ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) ) |
48 |
4 47
|
sylbi |
⊢ ( ω ≈ 𝑡 → ( 𝑡 ⊆ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) ) |
49 |
48
|
imp |
⊢ ( ( ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
50 |
49
|
exlimiv |
⊢ ( ∃ 𝑡 ( ω ≈ 𝑡 ∧ 𝑡 ⊆ 𝐴 ) → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |
51 |
3 50
|
syl6bi |
⊢ ( 𝐴 ∈ V → ( ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) ) |
52 |
2 51
|
mpcom |
⊢ ( ω ≼ 𝐴 → ¬ 𝒫 𝐴 ≼ ∪ 𝑛 ∈ ω ( 𝐴 ↑m 𝑛 ) ) |