Step |
Hyp |
Ref |
Expression |
1 |
|
sucelon |
⊢ ( 𝐴 ∈ On ↔ suc 𝐴 ∈ On ) |
2 |
|
cfval |
⊢ ( suc 𝐴 ∈ On → ( cf ‘ suc 𝐴 ) = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
3 |
1 2
|
sylbi |
⊢ ( 𝐴 ∈ On → ( cf ‘ suc 𝐴 ) = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
4 |
|
cardsn |
⊢ ( 𝐴 ∈ On → ( card ‘ { 𝐴 } ) = 1o ) |
5 |
4
|
eqcomd |
⊢ ( 𝐴 ∈ On → 1o = ( card ‘ { 𝐴 } ) ) |
6 |
|
snidg |
⊢ ( 𝐴 ∈ On → 𝐴 ∈ { 𝐴 } ) |
7 |
|
elsuci |
⊢ ( 𝑧 ∈ suc 𝐴 → ( 𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴 ) ) |
8 |
|
onelss |
⊢ ( 𝐴 ∈ On → ( 𝑧 ∈ 𝐴 → 𝑧 ⊆ 𝐴 ) ) |
9 |
|
eqimss |
⊢ ( 𝑧 = 𝐴 → 𝑧 ⊆ 𝐴 ) |
10 |
9
|
a1i |
⊢ ( 𝐴 ∈ On → ( 𝑧 = 𝐴 → 𝑧 ⊆ 𝐴 ) ) |
11 |
8 10
|
jaod |
⊢ ( 𝐴 ∈ On → ( ( 𝑧 ∈ 𝐴 ∨ 𝑧 = 𝐴 ) → 𝑧 ⊆ 𝐴 ) ) |
12 |
7 11
|
syl5 |
⊢ ( 𝐴 ∈ On → ( 𝑧 ∈ suc 𝐴 → 𝑧 ⊆ 𝐴 ) ) |
13 |
|
sseq2 |
⊢ ( 𝑤 = 𝐴 → ( 𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝐴 ) ) |
14 |
13
|
rspcev |
⊢ ( ( 𝐴 ∈ { 𝐴 } ∧ 𝑧 ⊆ 𝐴 ) → ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) |
15 |
6 12 14
|
syl6an |
⊢ ( 𝐴 ∈ On → ( 𝑧 ∈ suc 𝐴 → ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) ) |
16 |
15
|
ralrimiv |
⊢ ( 𝐴 ∈ On → ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) |
17 |
|
ssun2 |
⊢ { 𝐴 } ⊆ ( 𝐴 ∪ { 𝐴 } ) |
18 |
|
df-suc |
⊢ suc 𝐴 = ( 𝐴 ∪ { 𝐴 } ) |
19 |
17 18
|
sseqtrri |
⊢ { 𝐴 } ⊆ suc 𝐴 |
20 |
16 19
|
jctil |
⊢ ( 𝐴 ∈ On → ( { 𝐴 } ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) ) |
21 |
|
snex |
⊢ { 𝐴 } ∈ V |
22 |
|
fveq2 |
⊢ ( 𝑦 = { 𝐴 } → ( card ‘ 𝑦 ) = ( card ‘ { 𝐴 } ) ) |
23 |
22
|
eqeq2d |
⊢ ( 𝑦 = { 𝐴 } → ( 1o = ( card ‘ 𝑦 ) ↔ 1o = ( card ‘ { 𝐴 } ) ) ) |
24 |
|
sseq1 |
⊢ ( 𝑦 = { 𝐴 } → ( 𝑦 ⊆ suc 𝐴 ↔ { 𝐴 } ⊆ suc 𝐴 ) ) |
25 |
|
rexeq |
⊢ ( 𝑦 = { 𝐴 } → ( ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) ) |
26 |
25
|
ralbidv |
⊢ ( 𝑦 = { 𝐴 } → ( ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) ) |
27 |
24 26
|
anbi12d |
⊢ ( 𝑦 = { 𝐴 } → ( ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ↔ ( { 𝐴 } ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) ) ) |
28 |
23 27
|
anbi12d |
⊢ ( 𝑦 = { 𝐴 } → ( ( 1o = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ( 1o = ( card ‘ { 𝐴 } ) ∧ ( { 𝐴 } ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) ) ) ) |
29 |
21 28
|
spcev |
⊢ ( ( 1o = ( card ‘ { 𝐴 } ) ∧ ( { 𝐴 } ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ { 𝐴 } 𝑧 ⊆ 𝑤 ) ) → ∃ 𝑦 ( 1o = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
30 |
5 20 29
|
syl2anc |
⊢ ( 𝐴 ∈ On → ∃ 𝑦 ( 1o = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
31 |
|
1oex |
⊢ 1o ∈ V |
32 |
|
eqeq1 |
⊢ ( 𝑥 = 1o → ( 𝑥 = ( card ‘ 𝑦 ) ↔ 1o = ( card ‘ 𝑦 ) ) ) |
33 |
32
|
anbi1d |
⊢ ( 𝑥 = 1o → ( ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ( 1o = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
34 |
33
|
exbidv |
⊢ ( 𝑥 = 1o → ( ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ∃ 𝑦 ( 1o = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
35 |
31 34
|
elab |
⊢ ( 1o ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ↔ ∃ 𝑦 ( 1o = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
36 |
30 35
|
sylibr |
⊢ ( 𝐴 ∈ On → 1o ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
37 |
|
el1o |
⊢ ( 𝑣 ∈ 1o ↔ 𝑣 = ∅ ) |
38 |
|
eqcom |
⊢ ( ∅ = ( card ‘ 𝑦 ) ↔ ( card ‘ 𝑦 ) = ∅ ) |
39 |
|
vex |
⊢ 𝑦 ∈ V |
40 |
|
onssnum |
⊢ ( ( 𝑦 ∈ V ∧ 𝑦 ⊆ On ) → 𝑦 ∈ dom card ) |
41 |
39 40
|
mpan |
⊢ ( 𝑦 ⊆ On → 𝑦 ∈ dom card ) |
42 |
|
cardnueq0 |
⊢ ( 𝑦 ∈ dom card → ( ( card ‘ 𝑦 ) = ∅ ↔ 𝑦 = ∅ ) ) |
43 |
41 42
|
syl |
⊢ ( 𝑦 ⊆ On → ( ( card ‘ 𝑦 ) = ∅ ↔ 𝑦 = ∅ ) ) |
44 |
38 43
|
bitrid |
⊢ ( 𝑦 ⊆ On → ( ∅ = ( card ‘ 𝑦 ) ↔ 𝑦 = ∅ ) ) |
45 |
44
|
biimpa |
⊢ ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) → 𝑦 = ∅ ) |
46 |
|
rex0 |
⊢ ¬ ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 |
47 |
46
|
a1i |
⊢ ( 𝑧 ∈ suc 𝐴 → ¬ ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ) |
48 |
47
|
nrex |
⊢ ¬ ∃ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 |
49 |
|
nsuceq0 |
⊢ suc 𝐴 ≠ ∅ |
50 |
|
r19.2z |
⊢ ( ( suc 𝐴 ≠ ∅ ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ) → ∃ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ) |
51 |
49 50
|
mpan |
⊢ ( ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 → ∃ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ) |
52 |
48 51
|
mto |
⊢ ¬ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 |
53 |
|
rexeq |
⊢ ( 𝑦 = ∅ → ( ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ) ) |
54 |
53
|
ralbidv |
⊢ ( 𝑦 = ∅ → ( ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ ∅ 𝑧 ⊆ 𝑤 ) ) |
55 |
52 54
|
mtbiri |
⊢ ( 𝑦 = ∅ → ¬ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) |
56 |
45 55
|
syl |
⊢ ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) → ¬ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) |
57 |
56
|
intnand |
⊢ ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) → ¬ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) |
58 |
|
imnan |
⊢ ( ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) → ¬ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ¬ ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
59 |
57 58
|
mpbi |
⊢ ¬ ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) |
60 |
|
suceloni |
⊢ ( 𝐴 ∈ On → suc 𝐴 ∈ On ) |
61 |
|
onss |
⊢ ( suc 𝐴 ∈ On → suc 𝐴 ⊆ On ) |
62 |
|
sstr |
⊢ ( ( 𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ⊆ On ) → 𝑦 ⊆ On ) |
63 |
61 62
|
sylan2 |
⊢ ( ( 𝑦 ⊆ suc 𝐴 ∧ suc 𝐴 ∈ On ) → 𝑦 ⊆ On ) |
64 |
60 63
|
sylan2 |
⊢ ( ( 𝑦 ⊆ suc 𝐴 ∧ 𝐴 ∈ On ) → 𝑦 ⊆ On ) |
65 |
64
|
ancoms |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ suc 𝐴 ) → 𝑦 ⊆ On ) |
66 |
65
|
adantrr |
⊢ ( ( 𝐴 ∈ On ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → 𝑦 ⊆ On ) |
67 |
66
|
3adant2 |
⊢ ( ( 𝐴 ∈ On ∧ ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → 𝑦 ⊆ On ) |
68 |
|
simp2 |
⊢ ( ( 𝐴 ∈ On ∧ ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → ∅ = ( card ‘ 𝑦 ) ) |
69 |
|
simp3 |
⊢ ( ( 𝐴 ∈ On ∧ ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) |
70 |
67 68 69
|
jca31 |
⊢ ( ( 𝐴 ∈ On ∧ ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
71 |
70
|
3expib |
⊢ ( 𝐴 ∈ On → ( ( ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → ( ( 𝑦 ⊆ On ∧ ∅ = ( card ‘ 𝑦 ) ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
72 |
59 71
|
mtoi |
⊢ ( 𝐴 ∈ On → ¬ ( ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
73 |
72
|
nexdv |
⊢ ( 𝐴 ∈ On → ¬ ∃ 𝑦 ( ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
74 |
|
0ex |
⊢ ∅ ∈ V |
75 |
|
eqeq1 |
⊢ ( 𝑥 = ∅ → ( 𝑥 = ( card ‘ 𝑦 ) ↔ ∅ = ( card ‘ 𝑦 ) ) ) |
76 |
75
|
anbi1d |
⊢ ( 𝑥 = ∅ → ( ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ( ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
77 |
76
|
exbidv |
⊢ ( 𝑥 = ∅ → ( ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ∃ 𝑦 ( ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
78 |
74 77
|
elab |
⊢ ( ∅ ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ↔ ∃ 𝑦 ( ∅ = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
79 |
73 78
|
sylnibr |
⊢ ( 𝐴 ∈ On → ¬ ∅ ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
80 |
79
|
adantr |
⊢ ( ( 𝐴 ∈ On ∧ 𝑣 = ∅ ) → ¬ ∅ ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
81 |
|
eleq1 |
⊢ ( 𝑣 = ∅ → ( 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ↔ ∅ ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) ) |
82 |
81
|
adantl |
⊢ ( ( 𝐴 ∈ On ∧ 𝑣 = ∅ ) → ( 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ↔ ∅ ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) ) |
83 |
80 82
|
mtbird |
⊢ ( ( 𝐴 ∈ On ∧ 𝑣 = ∅ ) → ¬ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
84 |
37 83
|
sylan2b |
⊢ ( ( 𝐴 ∈ On ∧ 𝑣 ∈ 1o ) → ¬ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
85 |
84
|
ralrimiva |
⊢ ( 𝐴 ∈ On → ∀ 𝑣 ∈ 1o ¬ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
86 |
|
cardon |
⊢ ( card ‘ 𝑦 ) ∈ On |
87 |
|
eleq1 |
⊢ ( 𝑥 = ( card ‘ 𝑦 ) → ( 𝑥 ∈ On ↔ ( card ‘ 𝑦 ) ∈ On ) ) |
88 |
86 87
|
mpbiri |
⊢ ( 𝑥 = ( card ‘ 𝑦 ) → 𝑥 ∈ On ) |
89 |
88
|
adantr |
⊢ ( ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → 𝑥 ∈ On ) |
90 |
89
|
exlimiv |
⊢ ( ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → 𝑥 ∈ On ) |
91 |
90
|
abssi |
⊢ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ⊆ On |
92 |
|
oneqmini |
⊢ ( { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ⊆ On → ( ( 1o ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ∧ ∀ 𝑣 ∈ 1o ¬ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) → 1o = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) ) |
93 |
91 92
|
ax-mp |
⊢ ( ( 1o ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ∧ ∀ 𝑣 ∈ 1o ¬ 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) → 1o = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
94 |
36 85 93
|
syl2anc |
⊢ ( 𝐴 ∈ On → 1o = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ suc 𝐴 ∧ ∀ 𝑧 ∈ suc 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
95 |
3 94
|
eqtr4d |
⊢ ( 𝐴 ∈ On → ( cf ‘ suc 𝐴 ) = 1o ) |