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 |
|
simpl1 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) |
3 |
|
simpl2 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ) |
4 |
|
simprl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 ∈ 𝐵 ) |
5 |
|
simpl3 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) |
6 |
|
simp2l |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → 𝐵 ⊆ No ) |
7 |
6
|
sselda |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → 𝑤 ∈ No ) |
8 |
|
simp3l |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → 𝑈 ∈ 𝐵 ) |
9 |
6 8
|
sseldd |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → 𝑈 ∈ No ) |
10 |
9
|
adantr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → 𝑈 ∈ No ) |
11 |
|
sltso |
⊢ <s Or No |
12 |
|
soasym |
⊢ ( ( <s Or No ∧ ( 𝑤 ∈ No ∧ 𝑈 ∈ No ) ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) ) |
13 |
11 12
|
mpan |
⊢ ( ( 𝑤 ∈ No ∧ 𝑈 ∈ No ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) ) |
14 |
7 10 13
|
syl2anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) ) |
15 |
14
|
impr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ 𝑈 <s 𝑤 ) |
16 |
4 15
|
jca |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤 ) ) |
17 |
1
|
noinfbnd1lem2 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ∧ ( 𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤 ) ) ) → ( 𝑤 ↾ dom 𝑇 ) = 𝑇 ) |
18 |
2 3 5 16 17
|
syl112anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ↾ dom 𝑇 ) = 𝑇 ) |
19 |
1
|
noinfbnd1lem3 |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑤 ∈ 𝐵 ∧ ( 𝑤 ↾ dom 𝑇 ) = 𝑇 ) ) → ( 𝑤 ‘ dom 𝑇 ) ≠ 1o ) |
20 |
2 3 4 18 19
|
syl112anc |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ‘ dom 𝑇 ) ≠ 1o ) |
21 |
20
|
neneqd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ ( 𝑤 ‘ dom 𝑇 ) = 1o ) |
22 |
21
|
expr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ¬ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) |
23 |
|
imnan |
⊢ ( ( 𝑤 <s 𝑈 → ¬ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ↔ ¬ ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) |
24 |
22 23
|
sylib |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ 𝑤 ∈ 𝐵 ) → ¬ ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) |
25 |
24
|
nrexdv |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → ¬ ∃ 𝑤 ∈ 𝐵 ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) |
26 |
|
breq2 |
⊢ ( 𝑥 = 𝑈 → ( 𝑦 <s 𝑥 ↔ 𝑦 <s 𝑈 ) ) |
27 |
26
|
rexbidv |
⊢ ( 𝑥 = 𝑈 → ( ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑈 ) ) |
28 |
|
simpl1 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) |
29 |
|
dfral2 |
⊢ ( ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ¬ ∃ 𝑥 ∈ 𝐵 ¬ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ) |
30 |
|
ralnex |
⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ¬ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ) |
31 |
30
|
rexbii |
⊢ ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ↔ ∃ 𝑥 ∈ 𝐵 ¬ ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ) |
32 |
29 31
|
xchbinxr |
⊢ ( ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ↔ ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) |
33 |
28 32
|
sylibr |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∀ 𝑥 ∈ 𝐵 ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑥 ) |
34 |
|
simpl3l |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → 𝑈 ∈ 𝐵 ) |
35 |
27 33 34
|
rspcdva |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑈 ) |
36 |
|
breq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 <s 𝑈 ↔ 𝑤 <s 𝑈 ) ) |
37 |
36
|
cbvrexvw |
⊢ ( ∃ 𝑦 ∈ 𝐵 𝑦 <s 𝑈 ↔ ∃ 𝑤 ∈ 𝐵 𝑤 <s 𝑈 ) |
38 |
35 37
|
sylib |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∃ 𝑤 ∈ 𝐵 𝑤 <s 𝑈 ) |
39 |
|
simpl2l |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → 𝐵 ⊆ No ) |
40 |
39
|
adantr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝐵 ⊆ No ) |
41 |
|
simprl |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 ∈ 𝐵 ) |
42 |
40 41
|
sseldd |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 ∈ No ) |
43 |
34
|
adantr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑈 ∈ 𝐵 ) |
44 |
40 43
|
sseldd |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑈 ∈ No ) |
45 |
|
simpl2 |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ) |
46 |
1
|
noinfno |
⊢ ( ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) → 𝑇 ∈ No ) |
47 |
45 46
|
syl |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → 𝑇 ∈ No ) |
48 |
47
|
adantr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑇 ∈ No ) |
49 |
|
nodmon |
⊢ ( 𝑇 ∈ No → dom 𝑇 ∈ On ) |
50 |
48 49
|
syl |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → dom 𝑇 ∈ On ) |
51 |
|
simpll1 |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ) |
52 |
|
simpll2 |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ) |
53 |
|
simpll3 |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) |
54 |
|
simprr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → 𝑤 <s 𝑈 ) |
55 |
42 44 13
|
syl2anc |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 <s 𝑈 → ¬ 𝑈 <s 𝑤 ) ) |
56 |
54 55
|
mpd |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ¬ 𝑈 <s 𝑤 ) |
57 |
41 56
|
jca |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑤 ) ) |
58 |
51 52 53 57 17
|
syl112anc |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ↾ dom 𝑇 ) = 𝑇 ) |
59 |
|
simpl3r |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) |
60 |
59
|
adantr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) |
61 |
58 60
|
eqtr4d |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ↾ dom 𝑇 ) = ( 𝑈 ↾ dom 𝑇 ) ) |
62 |
|
simplr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑈 ‘ dom 𝑇 ) = ∅ ) |
63 |
|
nogt01o |
⊢ ( ( ( 𝑤 ∈ No ∧ 𝑈 ∈ No ∧ dom 𝑇 ∈ On ) ∧ ( ( 𝑤 ↾ dom 𝑇 ) = ( 𝑈 ↾ dom 𝑇 ) ∧ 𝑤 <s 𝑈 ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( 𝑤 ‘ dom 𝑇 ) = 1o ) |
64 |
42 44 50 61 54 62 63
|
syl321anc |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ ( 𝑤 ∈ 𝐵 ∧ 𝑤 <s 𝑈 ) ) → ( 𝑤 ‘ dom 𝑇 ) = 1o ) |
65 |
64
|
expr |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) |
66 |
65
|
ancld |
⊢ ( ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) ∧ 𝑤 ∈ 𝐵 ) → ( 𝑤 <s 𝑈 → ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) ) |
67 |
66
|
reximdva |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ( ∃ 𝑤 ∈ 𝐵 𝑤 <s 𝑈 → ∃ 𝑤 ∈ 𝐵 ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) ) |
68 |
38 67
|
mpd |
⊢ ( ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) ∧ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) → ∃ 𝑤 ∈ 𝐵 ( 𝑤 <s 𝑈 ∧ ( 𝑤 ‘ dom 𝑇 ) = 1o ) ) |
69 |
25 68
|
mtand |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → ¬ ( 𝑈 ‘ dom 𝑇 ) = ∅ ) |
70 |
69
|
neqned |
⊢ ( ( ¬ ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ ( 𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝑈 ∈ 𝐵 ∧ ( 𝑈 ↾ dom 𝑇 ) = 𝑇 ) ) → ( 𝑈 ‘ dom 𝑇 ) ≠ ∅ ) |