Step |
Hyp |
Ref |
Expression |
1 |
|
nosupbnd1.1 |
⊢ 𝑆 = if ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 , ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { 〈 dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o 〉 } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
2 |
|
simp2l |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → 𝐴 ⊆ No ) |
3 |
|
simp3 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → 𝑈 ∈ 𝐴 ) |
4 |
2 3
|
sseldd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → 𝑈 ∈ No ) |
5 |
1
|
nosupno |
⊢ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) → 𝑆 ∈ No ) |
6 |
5
|
3ad2ant2 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → 𝑆 ∈ No ) |
7 |
|
nodmon |
⊢ ( 𝑆 ∈ No → dom 𝑆 ∈ On ) |
8 |
6 7
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → dom 𝑆 ∈ On ) |
9 |
|
noreson |
⊢ ( ( 𝑈 ∈ No ∧ dom 𝑆 ∈ On ) → ( 𝑈 ↾ dom 𝑆 ) ∈ No ) |
10 |
4 8 9
|
syl2anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → ( 𝑈 ↾ dom 𝑆 ) ∈ No ) |
11 |
|
dmres |
⊢ dom ( 𝑈 ↾ dom 𝑆 ) = ( dom 𝑆 ∩ dom 𝑈 ) |
12 |
|
inss1 |
⊢ ( dom 𝑆 ∩ dom 𝑈 ) ⊆ dom 𝑆 |
13 |
11 12
|
eqsstri |
⊢ dom ( 𝑈 ↾ dom 𝑆 ) ⊆ dom 𝑆 |
14 |
13
|
a1i |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → dom ( 𝑈 ↾ dom 𝑆 ) ⊆ dom 𝑆 ) |
15 |
|
ssidd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → dom 𝑆 ⊆ dom 𝑆 ) |
16 |
|
iffalse |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → if ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 , ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { 〈 dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o 〉 } ) , ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
17 |
1 16
|
syl5eq |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → 𝑆 = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
18 |
17
|
dmeqd |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) ) |
19 |
|
iotaex |
⊢ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ∈ V |
20 |
|
eqid |
⊢ ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) = ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) |
21 |
19 20
|
dmmpti |
⊢ dom ( 𝑔 ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↦ ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) ) = { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } |
22 |
18 21
|
eqtrdi |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ) |
23 |
22
|
eleq2d |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ( ℎ ∈ dom 𝑆 ↔ ℎ ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ) ) |
24 |
|
vex |
⊢ ℎ ∈ V |
25 |
|
eleq1w |
⊢ ( 𝑦 = ℎ → ( 𝑦 ∈ dom 𝑢 ↔ ℎ ∈ dom 𝑢 ) ) |
26 |
|
suceq |
⊢ ( 𝑦 = ℎ → suc 𝑦 = suc ℎ ) |
27 |
26
|
reseq2d |
⊢ ( 𝑦 = ℎ → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑢 ↾ suc ℎ ) ) |
28 |
26
|
reseq2d |
⊢ ( 𝑦 = ℎ → ( 𝑣 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc ℎ ) ) |
29 |
27 28
|
eqeq12d |
⊢ ( 𝑦 = ℎ → ( ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ↔ ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) |
30 |
29
|
imbi2d |
⊢ ( 𝑦 = ℎ → ( ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
31 |
30
|
ralbidv |
⊢ ( 𝑦 = ℎ → ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
32 |
25 31
|
anbi12d |
⊢ ( 𝑦 = ℎ → ( ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ( ℎ ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
33 |
32
|
rexbidv |
⊢ ( 𝑦 = ℎ → ( ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑢 ∈ 𝐴 ( ℎ ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
34 |
|
dmeq |
⊢ ( 𝑢 = 𝑝 → dom 𝑢 = dom 𝑝 ) |
35 |
34
|
eleq2d |
⊢ ( 𝑢 = 𝑝 → ( ℎ ∈ dom 𝑢 ↔ ℎ ∈ dom 𝑝 ) ) |
36 |
|
breq2 |
⊢ ( 𝑢 = 𝑝 → ( 𝑣 <s 𝑢 ↔ 𝑣 <s 𝑝 ) ) |
37 |
36
|
notbid |
⊢ ( 𝑢 = 𝑝 → ( ¬ 𝑣 <s 𝑢 ↔ ¬ 𝑣 <s 𝑝 ) ) |
38 |
|
reseq1 |
⊢ ( 𝑢 = 𝑝 → ( 𝑢 ↾ suc ℎ ) = ( 𝑝 ↾ suc ℎ ) ) |
39 |
38
|
eqeq1d |
⊢ ( 𝑢 = 𝑝 → ( ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ↔ ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) |
40 |
37 39
|
imbi12d |
⊢ ( 𝑢 = 𝑝 → ( ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ↔ ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
41 |
40
|
ralbidv |
⊢ ( 𝑢 = 𝑝 → ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ↔ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
42 |
35 41
|
anbi12d |
⊢ ( 𝑢 = 𝑝 → ( ( ℎ ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ↔ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
43 |
42
|
cbvrexvw |
⊢ ( ∃ 𝑢 ∈ 𝐴 ( ℎ ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ↔ ∃ 𝑝 ∈ 𝐴 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
44 |
33 43
|
bitrdi |
⊢ ( 𝑦 = ℎ → ( ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) ↔ ∃ 𝑝 ∈ 𝐴 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
45 |
24 44
|
elab |
⊢ ( ℎ ∈ { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } ↔ ∃ 𝑝 ∈ 𝐴 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) |
46 |
23 45
|
bitrdi |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ( ℎ ∈ dom 𝑆 ↔ ∃ 𝑝 ∈ 𝐴 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
47 |
46
|
3ad2ant1 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → ( ℎ ∈ dom 𝑆 ↔ ∃ 𝑝 ∈ 𝐴 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) |
48 |
|
simpl1 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
49 |
|
simpl2 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ) |
50 |
|
simprl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑝 ∈ 𝐴 ) |
51 |
|
simprrl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ℎ ∈ dom 𝑝 ) |
52 |
|
simprrr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) |
53 |
1
|
nosupres |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑝 ∈ 𝐴 ∧ ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) → ( 𝑆 ↾ suc ℎ ) = ( 𝑝 ↾ suc ℎ ) ) |
54 |
48 49 50 51 52 53
|
syl113anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑆 ↾ suc ℎ ) = ( 𝑝 ↾ suc ℎ ) ) |
55 |
|
simpl2l |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝐴 ⊆ No ) |
56 |
55 50
|
sseldd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑝 ∈ No ) |
57 |
4
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑈 ∈ No ) |
58 |
|
sltso |
⊢ <s Or No |
59 |
|
soasym |
⊢ ( ( <s Or No ∧ ( 𝑝 ∈ No ∧ 𝑈 ∈ No ) ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) ) |
60 |
58 59
|
mpan |
⊢ ( ( 𝑝 ∈ No ∧ 𝑈 ∈ No ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) ) |
61 |
56 57 60
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑝 <s 𝑈 → ¬ 𝑈 <s 𝑝 ) ) |
62 |
|
simpl3 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → 𝑈 ∈ 𝐴 ) |
63 |
|
breq1 |
⊢ ( 𝑣 = 𝑈 → ( 𝑣 <s 𝑝 ↔ 𝑈 <s 𝑝 ) ) |
64 |
63
|
notbid |
⊢ ( 𝑣 = 𝑈 → ( ¬ 𝑣 <s 𝑝 ↔ ¬ 𝑈 <s 𝑝 ) ) |
65 |
|
reseq1 |
⊢ ( 𝑣 = 𝑈 → ( 𝑣 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) |
66 |
65
|
eqeq2d |
⊢ ( 𝑣 = 𝑈 → ( ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ↔ ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) |
67 |
64 66
|
imbi12d |
⊢ ( 𝑣 = 𝑈 → ( ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ↔ ( ¬ 𝑈 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) ) |
68 |
67
|
rspcv |
⊢ ( 𝑈 ∈ 𝐴 → ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) → ( ¬ 𝑈 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) ) |
69 |
62 52 68
|
sylc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( ¬ 𝑈 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) |
70 |
61 69
|
syld |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑝 <s 𝑈 → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) ) |
71 |
70
|
imp |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) ∧ 𝑝 <s 𝑈 ) → ( 𝑝 ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) |
72 |
|
nodmon |
⊢ ( 𝑝 ∈ No → dom 𝑝 ∈ On ) |
73 |
56 72
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → dom 𝑝 ∈ On ) |
74 |
|
onelon |
⊢ ( ( dom 𝑝 ∈ On ∧ ℎ ∈ dom 𝑝 ) → ℎ ∈ On ) |
75 |
73 51 74
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ℎ ∈ On ) |
76 |
|
sucelon |
⊢ ( ℎ ∈ On ↔ suc ℎ ∈ On ) |
77 |
75 76
|
sylib |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → suc ℎ ∈ On ) |
78 |
|
noreson |
⊢ ( ( 𝑈 ∈ No ∧ suc ℎ ∈ On ) → ( 𝑈 ↾ suc ℎ ) ∈ No ) |
79 |
57 77 78
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑈 ↾ suc ℎ ) ∈ No ) |
80 |
|
sonr |
⊢ ( ( <s Or No ∧ ( 𝑈 ↾ suc ℎ ) ∈ No ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
81 |
58 80
|
mpan |
⊢ ( ( 𝑈 ↾ suc ℎ ) ∈ No → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
82 |
79 81
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
83 |
82
|
adantr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) ∧ 𝑝 <s 𝑈 ) → ¬ ( 𝑈 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
84 |
71 83
|
eqnbrtrd |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) ∧ 𝑝 <s 𝑈 ) → ¬ ( 𝑝 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
85 |
84
|
ex |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( 𝑝 <s 𝑈 → ¬ ( 𝑝 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) ) |
86 |
|
sltres |
⊢ ( ( 𝑝 ∈ No ∧ 𝑈 ∈ No ∧ suc ℎ ∈ On ) → ( ( 𝑝 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) → 𝑝 <s 𝑈 ) ) |
87 |
56 57 77 86
|
syl3anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( ( 𝑝 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) → 𝑝 <s 𝑈 ) ) |
88 |
87
|
con3d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ( ¬ 𝑝 <s 𝑈 → ¬ ( 𝑝 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) ) |
89 |
85 88
|
pm2.61d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ( 𝑝 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
90 |
54 89
|
eqnbrtrd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) ) ) → ¬ ( 𝑆 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
91 |
90
|
rexlimdvaa |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → ( ∃ 𝑝 ∈ 𝐴 ( ℎ ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc ℎ ) = ( 𝑣 ↾ suc ℎ ) ) ) → ¬ ( 𝑆 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) ) |
92 |
47 91
|
sylbid |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → ( ℎ ∈ dom 𝑆 → ¬ ( 𝑆 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) ) |
93 |
92
|
imp |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ℎ ∈ dom 𝑆 ) → ¬ ( 𝑆 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) |
94 |
|
nodmord |
⊢ ( 𝑆 ∈ No → Ord dom 𝑆 ) |
95 |
|
ordsucss |
⊢ ( Ord dom 𝑆 → ( ℎ ∈ dom 𝑆 → suc ℎ ⊆ dom 𝑆 ) ) |
96 |
6 94 95
|
3syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → ( ℎ ∈ dom 𝑆 → suc ℎ ⊆ dom 𝑆 ) ) |
97 |
96
|
imp |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ℎ ∈ dom 𝑆 ) → suc ℎ ⊆ dom 𝑆 ) |
98 |
97
|
resabs1d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ℎ ∈ dom 𝑆 ) → ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ℎ ) = ( 𝑈 ↾ suc ℎ ) ) |
99 |
98
|
breq2d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ℎ ∈ dom 𝑆 ) → ( ( 𝑆 ↾ suc ℎ ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ℎ ) ↔ ( 𝑆 ↾ suc ℎ ) <s ( 𝑈 ↾ suc ℎ ) ) ) |
100 |
93 99
|
mtbird |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) ∧ ℎ ∈ dom 𝑆 ) → ¬ ( 𝑆 ↾ suc ℎ ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ℎ ) ) |
101 |
100
|
ralrimiva |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → ∀ ℎ ∈ dom 𝑆 ¬ ( 𝑆 ↾ suc ℎ ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ℎ ) ) |
102 |
|
noresle |
⊢ ( ( ( ( 𝑈 ↾ dom 𝑆 ) ∈ No ∧ 𝑆 ∈ No ) ∧ ( dom ( 𝑈 ↾ dom 𝑆 ) ⊆ dom 𝑆 ∧ dom 𝑆 ⊆ dom 𝑆 ∧ ∀ ℎ ∈ dom 𝑆 ¬ ( 𝑆 ↾ suc ℎ ) <s ( ( 𝑈 ↾ dom 𝑆 ) ↾ suc ℎ ) ) ) → ¬ 𝑆 <s ( 𝑈 ↾ dom 𝑆 ) ) |
103 |
10 6 14 15 101 102
|
syl23anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ 𝑈 ∈ 𝐴 ) → ¬ 𝑆 <s ( 𝑈 ↾ dom 𝑆 ) ) |