| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nosupbnd2.1 |
⊢ 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 |
|
nfv |
⊢ Ⅎ 𝑥 ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) |
| 3 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑍 |
| 4 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 |
| 5 |
|
nfriota1 |
⊢ Ⅎ 𝑥 ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 6 |
5
|
nfdm |
⊢ Ⅎ 𝑥 dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 7 |
|
nfcv |
⊢ Ⅎ 𝑥 2o |
| 8 |
6 7
|
nfop |
⊢ Ⅎ 𝑥 ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ |
| 9 |
8
|
nfsn |
⊢ Ⅎ 𝑥 { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } |
| 10 |
5 9
|
nfun |
⊢ Ⅎ 𝑥 ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) |
| 11 |
|
nfcv |
⊢ Ⅎ 𝑥 { 𝑦 ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑦 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑦 ) = ( 𝑣 ↾ suc 𝑦 ) ) ) } |
| 12 |
|
nfiota1 |
⊢ Ⅎ 𝑥 ( ℩ 𝑥 ∃ 𝑢 ∈ 𝐴 ( 𝑔 ∈ dom 𝑢 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑢 → ( 𝑢 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ∧ ( 𝑢 ‘ 𝑔 ) = 𝑥 ) ) |
| 13 |
11 12
|
nfmpt |
⊢ Ⅎ 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 ) ) ) |
| 14 |
4 10 13
|
nfif |
⊢ Ⅎ 𝑥 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 ) ) ) ) |
| 15 |
1 14
|
nfcxfr |
⊢ Ⅎ 𝑥 𝑆 |
| 16 |
15
|
nfdm |
⊢ Ⅎ 𝑥 dom 𝑆 |
| 17 |
3 16
|
nfres |
⊢ Ⅎ 𝑥 ( 𝑍 ↾ dom 𝑆 ) |
| 18 |
|
nfcv |
⊢ Ⅎ 𝑥 <s |
| 19 |
17 18 15
|
nfbr |
⊢ Ⅎ 𝑥 ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 |
| 20 |
19
|
nfn |
⊢ Ⅎ 𝑥 ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 |
| 21 |
2 20
|
nfim |
⊢ Ⅎ 𝑥 ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) |
| 22 |
|
simpl |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) |
| 23 |
|
rspe |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 24 |
23
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 25 |
|
nomaxmo |
⊢ ( 𝐴 ⊆ No → ∃* 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 26 |
25
|
3ad2ant1 |
⊢ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → ∃* 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 27 |
26
|
ad2antrl |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ∃* 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 28 |
|
reu5 |
⊢ ( ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ↔ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ∃* 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) |
| 29 |
24 27 28
|
sylanbrc |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 30 |
|
riota1 |
⊢ ( ∃! 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ↔ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 ) ) |
| 31 |
29 30
|
syl |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ↔ ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 ) ) |
| 32 |
22 31
|
mpbid |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 ) |
| 33 |
|
nosupbnd2lem1 |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) → ¬ ( 𝑍 ↾ suc dom 𝑥 ) <s ( 𝑥 ∪ { ⟨ dom 𝑥 , 2o ⟩ } ) ) |
| 34 |
33
|
3expb |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ¬ ( 𝑍 ↾ suc dom 𝑥 ) <s ( 𝑥 ∪ { ⟨ dom 𝑥 , 2o ⟩ } ) ) |
| 35 |
|
dmeq |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = dom 𝑥 ) |
| 36 |
35
|
suceqd |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = suc dom 𝑥 ) |
| 37 |
36
|
reseq2d |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) = ( 𝑍 ↾ suc dom 𝑥 ) ) |
| 38 |
|
id |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 ) |
| 39 |
35
|
opeq1d |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ = ⟨ dom 𝑥 , 2o ⟩ ) |
| 40 |
39
|
sneqd |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } = { ⟨ dom 𝑥 , 2o ⟩ } ) |
| 41 |
38 40
|
uneq12d |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) = ( 𝑥 ∪ { ⟨ dom 𝑥 , 2o ⟩ } ) ) |
| 42 |
37 41
|
breq12d |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → ( ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) <s ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ↔ ( 𝑍 ↾ suc dom 𝑥 ) <s ( 𝑥 ∪ { ⟨ dom 𝑥 , 2o ⟩ } ) ) ) |
| 43 |
42
|
notbid |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → ( ¬ ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) <s ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ↔ ¬ ( 𝑍 ↾ suc dom 𝑥 ) <s ( 𝑥 ∪ { ⟨ dom 𝑥 , 2o ⟩ } ) ) ) |
| 44 |
34 43
|
syl5ibrcom |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = 𝑥 → ¬ ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) <s ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) ) |
| 45 |
32 44
|
mpd |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ¬ ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) <s ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) |
| 46 |
|
iftrue |
⊢ ( ∃ 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 ) ) ) ) = ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) |
| 47 |
1 46
|
eqtrid |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → 𝑆 = ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) |
| 48 |
23 47
|
syl |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) → 𝑆 = ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) |
| 49 |
48
|
dmeqd |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) → dom 𝑆 = dom ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) |
| 50 |
|
2on |
⊢ 2o ∈ On |
| 51 |
50
|
elexi |
⊢ 2o ∈ V |
| 52 |
51
|
dmsnop |
⊢ dom { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } = { dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) } |
| 53 |
52
|
uneq2i |
⊢ ( dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ dom { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) = ( dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) } ) |
| 54 |
|
dmun |
⊢ dom ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) = ( dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ dom { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) |
| 55 |
|
df-suc |
⊢ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) = ( dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) } ) |
| 56 |
53 54 55
|
3eqtr4i |
⊢ dom ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) = suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 57 |
49 56
|
eqtrdi |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) → dom 𝑆 = suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) |
| 58 |
57
|
reseq2d |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) → ( 𝑍 ↾ dom 𝑆 ) = ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) ) |
| 59 |
58
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝑍 ↾ dom 𝑆 ) = ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) ) |
| 60 |
48
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → 𝑆 = ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) |
| 61 |
59 60
|
breq12d |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ↔ ( 𝑍 ↾ suc dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ) <s ( ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∪ { ⟨ dom ( ℩ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) , 2o ⟩ } ) ) ) |
| 62 |
45 61
|
mtbird |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) |
| 63 |
62
|
exp31 |
⊢ ( 𝑥 ∈ 𝐴 → ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ) ) |
| 64 |
21 63
|
rexlimi |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ) |
| 65 |
64
|
imp |
⊢ ( ( ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) |
| 66 |
1
|
nosupno |
⊢ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) → 𝑆 ∈ No ) |
| 67 |
66
|
3adant3 |
⊢ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → 𝑆 ∈ No ) |
| 68 |
67
|
ad2antrl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → 𝑆 ∈ No ) |
| 69 |
|
nodmon |
⊢ ( 𝑆 ∈ No → dom 𝑆 ∈ On ) |
| 70 |
68 69
|
syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → dom 𝑆 ∈ On ) |
| 71 |
|
noreson |
⊢ ( ( 𝑆 ∈ No ∧ dom 𝑆 ∈ On ) → ( 𝑆 ↾ dom 𝑆 ) ∈ No ) |
| 72 |
68 70 71
|
syl2anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝑆 ↾ dom 𝑆 ) ∈ No ) |
| 73 |
|
simprl3 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → 𝑍 ∈ No ) |
| 74 |
|
noreson |
⊢ ( ( 𝑍 ∈ No ∧ dom 𝑆 ∈ On ) → ( 𝑍 ↾ dom 𝑆 ) ∈ No ) |
| 75 |
73 70 74
|
syl2anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝑍 ↾ dom 𝑆 ) ∈ No ) |
| 76 |
|
dmres |
⊢ dom ( 𝑆 ↾ dom 𝑆 ) = ( dom 𝑆 ∩ dom 𝑆 ) |
| 77 |
|
inss2 |
⊢ ( dom 𝑆 ∩ dom 𝑆 ) ⊆ dom 𝑆 |
| 78 |
76 77
|
eqsstri |
⊢ dom ( 𝑆 ↾ dom 𝑆 ) ⊆ dom 𝑆 |
| 79 |
78
|
a1i |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → dom ( 𝑆 ↾ dom 𝑆 ) ⊆ dom 𝑆 ) |
| 80 |
|
dmres |
⊢ dom ( 𝑍 ↾ dom 𝑆 ) = ( dom 𝑆 ∩ dom 𝑍 ) |
| 81 |
|
inss1 |
⊢ ( dom 𝑆 ∩ dom 𝑍 ) ⊆ dom 𝑆 |
| 82 |
80 81
|
eqsstri |
⊢ dom ( 𝑍 ↾ dom 𝑆 ) ⊆ dom 𝑆 |
| 83 |
82
|
a1i |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → dom ( 𝑍 ↾ dom 𝑆 ) ⊆ dom 𝑆 ) |
| 84 |
1
|
nosupdm |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = { 𝑔 ∣ ∃ 𝑝 ∈ 𝐴 ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) } ) |
| 85 |
84
|
eqabrd |
⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → ( 𝑔 ∈ dom 𝑆 ↔ ∃ 𝑝 ∈ 𝐴 ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) |
| 86 |
85
|
adantr |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝑔 ∈ dom 𝑆 ↔ ∃ 𝑝 ∈ 𝐴 ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) |
| 87 |
|
simprl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → 𝑝 ∈ 𝐴 ) |
| 88 |
|
simplrr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) |
| 89 |
|
breq1 |
⊢ ( 𝑎 = 𝑝 → ( 𝑎 <s 𝑍 ↔ 𝑝 <s 𝑍 ) ) |
| 90 |
89
|
rspcv |
⊢ ( 𝑝 ∈ 𝐴 → ( ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 → 𝑝 <s 𝑍 ) ) |
| 91 |
87 88 90
|
sylc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → 𝑝 <s 𝑍 ) |
| 92 |
|
simprl1 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → 𝐴 ⊆ No ) |
| 93 |
92
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → 𝐴 ⊆ No ) |
| 94 |
93 87
|
sseldd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → 𝑝 ∈ No ) |
| 95 |
73
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → 𝑍 ∈ No ) |
| 96 |
|
ltsso |
⊢ <s Or No |
| 97 |
|
soasym |
⊢ ( ( <s Or No ∧ ( 𝑝 ∈ No ∧ 𝑍 ∈ No ) ) → ( 𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝 ) ) |
| 98 |
96 97
|
mpan |
⊢ ( ( 𝑝 ∈ No ∧ 𝑍 ∈ No ) → ( 𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝 ) ) |
| 99 |
94 95 98
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ( 𝑝 <s 𝑍 → ¬ 𝑍 <s 𝑝 ) ) |
| 100 |
91 99
|
mpd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ¬ 𝑍 <s 𝑝 ) |
| 101 |
|
nodmon |
⊢ ( 𝑝 ∈ No → dom 𝑝 ∈ On ) |
| 102 |
94 101
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → dom 𝑝 ∈ On ) |
| 103 |
|
simprrl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → 𝑔 ∈ dom 𝑝 ) |
| 104 |
|
onelon |
⊢ ( ( dom 𝑝 ∈ On ∧ 𝑔 ∈ dom 𝑝 ) → 𝑔 ∈ On ) |
| 105 |
102 103 104
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → 𝑔 ∈ On ) |
| 106 |
|
onsucb |
⊢ ( 𝑔 ∈ On ↔ suc 𝑔 ∈ On ) |
| 107 |
105 106
|
sylib |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → suc 𝑔 ∈ On ) |
| 108 |
|
ltsres |
⊢ ( ( 𝑍 ∈ No ∧ 𝑝 ∈ No ∧ suc 𝑔 ∈ On ) → ( ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑝 ↾ suc 𝑔 ) → 𝑍 <s 𝑝 ) ) |
| 109 |
95 94 107 108
|
syl3anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ( ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑝 ↾ suc 𝑔 ) → 𝑍 <s 𝑝 ) ) |
| 110 |
100 109
|
mtod |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ¬ ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑝 ↾ suc 𝑔 ) ) |
| 111 |
|
simpll |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ) |
| 112 |
|
simprl2 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → 𝐴 ∈ V ) |
| 113 |
92 112
|
jca |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ) |
| 114 |
113
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ) |
| 115 |
|
simprrr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) |
| 116 |
|
breq1 |
⊢ ( 𝑣 = 𝑞 → ( 𝑣 <s 𝑝 ↔ 𝑞 <s 𝑝 ) ) |
| 117 |
116
|
notbid |
⊢ ( 𝑣 = 𝑞 → ( ¬ 𝑣 <s 𝑝 ↔ ¬ 𝑞 <s 𝑝 ) ) |
| 118 |
|
reseq1 |
⊢ ( 𝑣 = 𝑞 → ( 𝑣 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) |
| 119 |
118
|
eqeq2d |
⊢ ( 𝑣 = 𝑞 → ( ( 𝑝 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ↔ ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) |
| 120 |
117 119
|
imbi12d |
⊢ ( 𝑣 = 𝑞 → ( ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ↔ ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) |
| 121 |
120
|
cbvralvw |
⊢ ( ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ↔ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) |
| 122 |
115 121
|
sylibr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ) |
| 123 |
1
|
nosupres |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ) ∧ ( 𝑝 ∈ 𝐴 ∧ 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑣 ∈ 𝐴 ( ¬ 𝑣 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑣 ↾ suc 𝑔 ) ) ) ) → ( 𝑆 ↾ suc 𝑔 ) = ( 𝑝 ↾ suc 𝑔 ) ) |
| 124 |
111 114 87 103 122 123
|
syl113anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ( 𝑆 ↾ suc 𝑔 ) = ( 𝑝 ↾ suc 𝑔 ) ) |
| 125 |
124
|
breq2d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ( ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑆 ↾ suc 𝑔 ) ↔ ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑝 ↾ suc 𝑔 ) ) ) |
| 126 |
110 125
|
mtbird |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ ( 𝑝 ∈ 𝐴 ∧ ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) ) ) → ¬ ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑆 ↾ suc 𝑔 ) ) |
| 127 |
126
|
rexlimdvaa |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( ∃ 𝑝 ∈ 𝐴 ( 𝑔 ∈ dom 𝑝 ∧ ∀ 𝑞 ∈ 𝐴 ( ¬ 𝑞 <s 𝑝 → ( 𝑝 ↾ suc 𝑔 ) = ( 𝑞 ↾ suc 𝑔 ) ) ) → ¬ ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑆 ↾ suc 𝑔 ) ) ) |
| 128 |
86 127
|
sylbid |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝑔 ∈ dom 𝑆 → ¬ ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑆 ↾ suc 𝑔 ) ) ) |
| 129 |
128
|
imp |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → ¬ ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑆 ↾ suc 𝑔 ) ) |
| 130 |
68
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → 𝑆 ∈ No ) |
| 131 |
|
nodmord |
⊢ ( 𝑆 ∈ No → Ord dom 𝑆 ) |
| 132 |
130 131
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → Ord dom 𝑆 ) |
| 133 |
|
simpr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → 𝑔 ∈ dom 𝑆 ) |
| 134 |
|
ordsucss |
⊢ ( Ord dom 𝑆 → ( 𝑔 ∈ dom 𝑆 → suc 𝑔 ⊆ dom 𝑆 ) ) |
| 135 |
132 133 134
|
sylc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → suc 𝑔 ⊆ dom 𝑆 ) |
| 136 |
135
|
resabs1d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → ( ( 𝑍 ↾ dom 𝑆 ) ↾ suc 𝑔 ) = ( 𝑍 ↾ suc 𝑔 ) ) |
| 137 |
135
|
resabs1d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → ( ( 𝑆 ↾ dom 𝑆 ) ↾ suc 𝑔 ) = ( 𝑆 ↾ suc 𝑔 ) ) |
| 138 |
136 137
|
breq12d |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → ( ( ( 𝑍 ↾ dom 𝑆 ) ↾ suc 𝑔 ) <s ( ( 𝑆 ↾ dom 𝑆 ) ↾ suc 𝑔 ) ↔ ( 𝑍 ↾ suc 𝑔 ) <s ( 𝑆 ↾ suc 𝑔 ) ) ) |
| 139 |
129 138
|
mtbird |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) ∧ 𝑔 ∈ dom 𝑆 ) → ¬ ( ( 𝑍 ↾ dom 𝑆 ) ↾ suc 𝑔 ) <s ( ( 𝑆 ↾ dom 𝑆 ) ↾ suc 𝑔 ) ) |
| 140 |
139
|
ralrimiva |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ∀ 𝑔 ∈ dom 𝑆 ¬ ( ( 𝑍 ↾ dom 𝑆 ) ↾ suc 𝑔 ) <s ( ( 𝑆 ↾ dom 𝑆 ) ↾ suc 𝑔 ) ) |
| 141 |
|
noresle |
⊢ ( ( ( ( 𝑆 ↾ dom 𝑆 ) ∈ No ∧ ( 𝑍 ↾ dom 𝑆 ) ∈ No ) ∧ ( dom ( 𝑆 ↾ dom 𝑆 ) ⊆ dom 𝑆 ∧ dom ( 𝑍 ↾ dom 𝑆 ) ⊆ dom 𝑆 ∧ ∀ 𝑔 ∈ dom 𝑆 ¬ ( ( 𝑍 ↾ dom 𝑆 ) ↾ suc 𝑔 ) <s ( ( 𝑆 ↾ dom 𝑆 ) ↾ suc 𝑔 ) ) ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s ( 𝑆 ↾ dom 𝑆 ) ) |
| 142 |
72 75 79 83 140 141
|
syl23anc |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s ( 𝑆 ↾ dom 𝑆 ) ) |
| 143 |
|
nofun |
⊢ ( 𝑆 ∈ No → Fun 𝑆 ) |
| 144 |
|
funrel |
⊢ ( Fun 𝑆 → Rel 𝑆 ) |
| 145 |
|
resdm |
⊢ ( Rel 𝑆 → ( 𝑆 ↾ dom 𝑆 ) = 𝑆 ) |
| 146 |
68 143 144 145
|
4syl |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( 𝑆 ↾ dom 𝑆 ) = 𝑆 ) |
| 147 |
146
|
breq2d |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ( ( 𝑍 ↾ dom 𝑆 ) <s ( 𝑆 ↾ dom 𝑆 ) ↔ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ) |
| 148 |
142 147
|
mtbid |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) |
| 149 |
65 148
|
pm2.61ian |
⊢ ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) |
| 150 |
|
simpll1 |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → 𝐴 ⊆ No ) |
| 151 |
|
simpll2 |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → 𝐴 ∈ V ) |
| 152 |
|
simpr |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ 𝐴 ) |
| 153 |
1
|
nosupbnd1 |
⊢ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑎 ∈ 𝐴 ) → ( 𝑎 ↾ dom 𝑆 ) <s 𝑆 ) |
| 154 |
150 151 152 153
|
syl3anc |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → ( 𝑎 ↾ dom 𝑆 ) <s 𝑆 ) |
| 155 |
|
simplr |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) |
| 156 |
|
simpl1 |
⊢ ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) → 𝐴 ⊆ No ) |
| 157 |
156
|
sselda |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → 𝑎 ∈ No ) |
| 158 |
150 151 66
|
syl2anc |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → 𝑆 ∈ No ) |
| 159 |
158 69
|
syl |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → dom 𝑆 ∈ On ) |
| 160 |
|
noreson |
⊢ ( ( 𝑎 ∈ No ∧ dom 𝑆 ∈ On ) → ( 𝑎 ↾ dom 𝑆 ) ∈ No ) |
| 161 |
157 159 160
|
syl2anc |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → ( 𝑎 ↾ dom 𝑆 ) ∈ No ) |
| 162 |
|
simpll3 |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → 𝑍 ∈ No ) |
| 163 |
162 159 74
|
syl2anc |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → ( 𝑍 ↾ dom 𝑆 ) ∈ No ) |
| 164 |
|
sotr3 |
⊢ ( ( <s Or No ∧ ( ( 𝑎 ↾ dom 𝑆 ) ∈ No ∧ 𝑆 ∈ No ∧ ( 𝑍 ↾ dom 𝑆 ) ∈ No ) ) → ( ( ( 𝑎 ↾ dom 𝑆 ) <s 𝑆 ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) → ( 𝑎 ↾ dom 𝑆 ) <s ( 𝑍 ↾ dom 𝑆 ) ) ) |
| 165 |
96 164
|
mpan |
⊢ ( ( ( 𝑎 ↾ dom 𝑆 ) ∈ No ∧ 𝑆 ∈ No ∧ ( 𝑍 ↾ dom 𝑆 ) ∈ No ) → ( ( ( 𝑎 ↾ dom 𝑆 ) <s 𝑆 ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) → ( 𝑎 ↾ dom 𝑆 ) <s ( 𝑍 ↾ dom 𝑆 ) ) ) |
| 166 |
161 158 163 165
|
syl3anc |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → ( ( ( 𝑎 ↾ dom 𝑆 ) <s 𝑆 ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) → ( 𝑎 ↾ dom 𝑆 ) <s ( 𝑍 ↾ dom 𝑆 ) ) ) |
| 167 |
154 155 166
|
mp2and |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → ( 𝑎 ↾ dom 𝑆 ) <s ( 𝑍 ↾ dom 𝑆 ) ) |
| 168 |
|
ltsres |
⊢ ( ( 𝑎 ∈ No ∧ 𝑍 ∈ No ∧ dom 𝑆 ∈ On ) → ( ( 𝑎 ↾ dom 𝑆 ) <s ( 𝑍 ↾ dom 𝑆 ) → 𝑎 <s 𝑍 ) ) |
| 169 |
157 162 159 168
|
syl3anc |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → ( ( 𝑎 ↾ dom 𝑆 ) <s ( 𝑍 ↾ dom 𝑆 ) → 𝑎 <s 𝑍 ) ) |
| 170 |
167 169
|
mpd |
⊢ ( ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ∧ 𝑎 ∈ 𝐴 ) → 𝑎 <s 𝑍 ) |
| 171 |
170
|
ralrimiva |
⊢ ( ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) → ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ) |
| 172 |
149 171
|
impbida |
⊢ ( ( 𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → ( ∀ 𝑎 ∈ 𝐴 𝑎 <s 𝑍 ↔ ¬ ( 𝑍 ↾ dom 𝑆 ) <s 𝑆 ) ) |