| Step |
Hyp |
Ref |
Expression |
| 1 |
|
noinfbnd1.1 |
⊢ 𝑇 = if ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 , ( ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) , 1o ⟩ } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐵 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐵 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑢 <s 𝑣 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
| 2 |
1
|
noinfno |
⊢ ( ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) → 𝑇 ∈ No ) |
| 3 |
2
|
3ad2ant2 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → 𝑇 ∈ No ) |
| 4 |
|
simp2l |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → 𝐵 ⊆ No ) |
| 5 |
|
simp3 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → 𝑈 ∈ 𝐵 ) |
| 6 |
4 5
|
sseldd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → 𝑈 ∈ No ) |
| 7 |
|
nodmon |
⊢ ( 𝑇 ∈ No → dom 𝑇 ∈ On ) |
| 8 |
3 7
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → dom 𝑇 ∈ On ) |
| 9 |
|
noreson |
⊢ ( ( 𝑈 ∈ No ∧ dom 𝑇 ∈ On ) → ( 𝑈 ↾ dom 𝑇 ) ∈ No ) |
| 10 |
6 8 9
|
syl2anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → ( 𝑈 ↾ dom 𝑇 ) ∈ No ) |
| 11 |
|
ssidd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → dom 𝑇 ⊆ dom 𝑇 ) |
| 12 |
|
dmres |
⊢ dom ( 𝑈 ↾ dom 𝑇 ) = ( dom 𝑇 ∩ dom 𝑈 ) |
| 13 |
|
inss1 |
⊢ ( dom 𝑇 ∩ dom 𝑈 ) ⊆ dom 𝑇 |
| 14 |
12 13
|
eqsstri |
⊢ dom ( 𝑈 ↾ dom 𝑇 ) ⊆ dom 𝑇 |
| 15 |
14
|
a1i |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → dom ( 𝑈 ↾ dom 𝑇 ) ⊆ dom 𝑇 ) |
| 16 |
|
nodmord |
⊢ ( 𝑇 ∈ No → Ord dom 𝑇 ) |
| 17 |
3 16
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → Ord dom 𝑇 ) |
| 18 |
|
ordsucss |
⊢ ( Ord dom 𝑇 → ( ℎ ∈ dom 𝑇 → suc ℎ ⊆ dom 𝑇 ) ) |
| 19 |
17 18
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → ( ℎ ∈ dom 𝑇 → suc ℎ ⊆ dom 𝑇 ) ) |
| 20 |
19
|
imp |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ℎ ∈ dom 𝑇 ) → suc ℎ ⊆ dom 𝑇 ) |
| 21 |
20
|
resabs1d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ℎ ∈ dom 𝑇 ) → ( ( 𝑈 ↾ dom 𝑇 ) ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) |
| 22 |
1
|
noinfdm |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = { ℎ ∣ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ) } ) |
| 23 |
22
|
eleq2d |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → ( ℎ ∈ dom 𝑇 ↔ ℎ ∈ { ℎ ∣ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ) } ) ) |
| 24 |
|
abid |
⊢ ( ℎ ∈ { ℎ ∣ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ) } ↔ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ) ) |
| 25 |
|
breq2 |
⊢ ( 𝑞 = 𝑣 → ( 𝑝 <s 𝑞 ↔ 𝑝 <s 𝑣 ) ) |
| 26 |
25
|
notbid |
⊢ ( 𝑞 = 𝑣 → ( ¬ 𝑝 <s 𝑞 ↔ ¬ 𝑝 <s 𝑣 ) ) |
| 27 |
|
reseq1 |
⊢ ( 𝑞 = 𝑣 → ( 𝑞 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) |
| 28 |
27
|
eqeq2d |
⊢ ( 𝑞 = 𝑣 → ( ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ↔ ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) |
| 29 |
26 28
|
imbi12d |
⊢ ( 𝑞 = 𝑣 → ( ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ↔ ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
| 30 |
29
|
cbvralvw |
⊢ ( ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ↔ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) |
| 31 |
30
|
anbi2i |
⊢ ( ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ) ↔ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
| 32 |
31
|
rexbii |
⊢ ( ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ) ↔ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
| 33 |
24 32
|
bitri |
⊢ ( ℎ ∈ { ℎ ∣ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐵 ( ¬ 𝑝 <s 𝑞 → ( 𝑝 ↾ suc ℎ ) = ( 𝑞 ↾ suc ℎ ) ) ) } ↔ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
| 34 |
23 33
|
bitrdi |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → ( ℎ ∈ dom 𝑇 ↔ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
| 35 |
34
|
3ad2ant1 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → ( ℎ ∈ dom 𝑇 ↔ ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
| 36 |
|
simpl2l |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝐵 ⊆ No ) |
| 37 |
|
simprl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑝 ∈ 𝐵 ) |
| 38 |
36 37
|
sseldd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑝 ∈ No ) |
| 39 |
6
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑈 ∈ No ) |
| 40 |
|
ltsso |
⊢ <s Or No |
| 41 |
|
soasym |
⊢ ( ( <s Or No ∧ ( 𝑝 ∈ No ∧ 𝑈 ∈ No ) ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) ) |
| 42 |
40 41
|
mpan |
⊢ ( ( 𝑝 ∈ No ∧ 𝑈 ∈ No ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) ) |
| 43 |
38 39 42
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) ) |
| 44 |
|
nodmon |
⊢ ( 𝑝 ∈ No → dom 𝑝 ∈ On ) |
| 45 |
38 44
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → dom 𝑝 ∈ On ) |
| 46 |
|
simprrl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ℎ ∈ dom 𝑝 ) |
| 47 |
|
onelon |
⊢ ( ( dom 𝑝 ∈ On ∧ ℎ ∈ dom 𝑝 ) → ℎ ∈ On ) |
| 48 |
45 46 47
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ℎ ∈ On ) |
| 49 |
|
onsucb |
⊢ ( ℎ ∈ On ↔ suc ℎ ∈ On ) |
| 50 |
48 49
|
sylib |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → suc ℎ ∈ On ) |
| 51 |
|
ltsres |
⊢ ( ( 𝑈 ∈ No ∧ 𝑝 ∈ No ∧ suc ℎ ∈ On ) → ( ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) → 𝑈 <s 𝑝 ) ) |
| 52 |
39 38 50 51
|
syl3anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) → 𝑈 <s 𝑝 ) ) |
| 53 |
43 52
|
nsyld |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑝 <s 𝑈 → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) ) ) |
| 54 |
|
noreson |
⊢ ( ( 𝑈 ∈ No ∧ suc ℎ ∈ On ) → ( 𝑈 ↾ suc ℎ ) ∈ No ) |
| 55 |
39 50 54
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑈 ↾ suc ℎ ) ∈ No ) |
| 56 |
|
sonr |
⊢ ( ( <s Or No ∧ ( 𝑈 ↾ suc ℎ ) ∈ No ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
| 57 |
40 56
|
mpan |
⊢ ( ( 𝑈 ↾ suc ℎ ) ∈ No → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
| 58 |
55 57
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
| 59 |
58
|
adantr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) ∧ ¬ 𝑝 <s 𝑈 ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
| 60 |
|
breq2 |
⊢ ( 𝑣 = 𝑈 → ( 𝑝 <s 𝑣 ↔ 𝑝 <s 𝑈 ) ) |
| 61 |
60
|
notbid |
⊢ ( 𝑣 = 𝑈 → ( ¬ 𝑝 <s 𝑣 ↔ ¬ 𝑝 <s 𝑈 ) ) |
| 62 |
|
reseq1 |
⊢ ( 𝑣 = 𝑈 → ( 𝑣 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) |
| 63 |
62
|
eqeq2d |
⊢ ( 𝑣 = 𝑈 → ( ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ↔ ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) |
| 64 |
61 63
|
imbi12d |
⊢ ( 𝑣 = 𝑈 → ( ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ↔ ( ¬ 𝑝 <s 𝑈 → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) ) |
| 65 |
|
simprrr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) |
| 66 |
|
simpl3 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑈 ∈ 𝐵 ) |
| 67 |
64 65 66
|
rspcdva |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( ¬ 𝑝 <s 𝑈 → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) |
| 68 |
67
|
imp |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) ∧ ¬ 𝑝 <s 𝑈 ) → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) |
| 69 |
68
|
breq2d |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) ∧ ¬ 𝑝 <s 𝑈 ) → ( ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) ↔ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) ) |
| 70 |
59 69
|
mtbird |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) ∧ ¬ 𝑝 <s 𝑈 ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) ) |
| 71 |
70
|
ex |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( ¬ 𝑝 <s 𝑈 → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) ) ) |
| 72 |
53 71
|
pm2.61d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) ) |
| 73 |
|
simpl1 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) |
| 74 |
|
simpl2 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ) |
| 75 |
1
|
noinfres |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) → ( 𝑇 ↾ suc ℎ ) = ( 𝑝 ↾ suc ℎ ) ) |
| 76 |
73 74 37 46 65 75
|
syl113anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑇 ↾ suc ℎ ) = ( 𝑝 ↾ suc ℎ ) ) |
| 77 |
76
|
breq2d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( ( 𝑈 ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ↔ ( 𝑈 ↾ suc ℎ ) <s ( 𝑝 ↾ suc ℎ ) ) ) |
| 78 |
72 77
|
mtbird |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ( 𝑝 ∈ 𝐵 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ) |
| 79 |
78
|
rexlimdvaa |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → ( ∃ 𝑝 ∈ 𝐵 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐵 ( ¬ 𝑝 <s 𝑣 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ) ) |
| 80 |
35 79
|
sylbid |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → ( ℎ ∈ dom 𝑇 → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ) ) |
| 81 |
80
|
imp |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ℎ ∈ dom 𝑇 ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ) |
| 82 |
21 81
|
eqnbrtrd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) ∧ ℎ ∈ dom 𝑇 ) → ¬ ( ( 𝑈 ↾ dom 𝑇 ) ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ) |
| 83 |
82
|
ralrimiva |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → ∀ ℎ ∈ dom 𝑇 ¬ ( ( 𝑈 ↾ dom 𝑇 ) ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ) |
| 84 |
|
noresle |
⊢ ( ( ( 𝑇 ∈ No ∧ ( 𝑈 ↾ dom 𝑇 ) ∈ No ) ∧ ( dom 𝑇 ⊆ dom 𝑇 ∧ dom ( 𝑈 ↾ dom 𝑇 ) ⊆ dom 𝑇 ∧ ∀ ℎ ∈ dom 𝑇 ¬ ( ( 𝑈 ↾ dom 𝑇 ) ↾ suc ℎ ) <s ( 𝑇 ↾ suc ℎ ) ) ) → ¬ ( 𝑈 ↾ dom 𝑇 ) <s 𝑇 ) |
| 85 |
3 10 11 15 83 84
|
syl23anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑈 ∈ 𝐵 ) → ¬ ( 𝑈 ↾ dom 𝑇 ) <s 𝑇 ) |