Step |
Hyp |
Ref |
Expression |
1 |
|
1n0 |
⊢ 1o ≠ ∅ |
2 |
|
neeq1 |
⊢ ( ( rank ‘ 𝐴 ) = 1o → ( ( rank ‘ 𝐴 ) ≠ ∅ ↔ 1o ≠ ∅ ) ) |
3 |
1 2
|
mpbiri |
⊢ ( ( rank ‘ 𝐴 ) = 1o → ( rank ‘ 𝐴 ) ≠ ∅ ) |
4 |
3
|
neneqd |
⊢ ( ( rank ‘ 𝐴 ) = 1o → ¬ ( rank ‘ 𝐴 ) = ∅ ) |
5 |
|
fvprc |
⊢ ( ¬ 𝐴 ∈ V → ( rank ‘ 𝐴 ) = ∅ ) |
6 |
4 5
|
nsyl2 |
⊢ ( ( rank ‘ 𝐴 ) = 1o → 𝐴 ∈ V ) |
7 |
|
fveqeq2 |
⊢ ( 𝑥 = 𝐴 → ( ( rank ‘ 𝑥 ) = 1o ↔ ( rank ‘ 𝐴 ) = 1o ) ) |
8 |
|
eqeq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 1o ↔ 𝐴 = 1o ) ) |
9 |
7 8
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( ( rank ‘ 𝑥 ) = 1o → 𝑥 = 1o ) ↔ ( ( rank ‘ 𝐴 ) = 1o → 𝐴 = 1o ) ) ) |
10 |
|
neeq1 |
⊢ ( ( rank ‘ 𝑥 ) = 1o → ( ( rank ‘ 𝑥 ) ≠ ∅ ↔ 1o ≠ ∅ ) ) |
11 |
1 10
|
mpbiri |
⊢ ( ( rank ‘ 𝑥 ) = 1o → ( rank ‘ 𝑥 ) ≠ ∅ ) |
12 |
|
vex |
⊢ 𝑥 ∈ V |
13 |
12
|
rankeq0 |
⊢ ( 𝑥 = ∅ ↔ ( rank ‘ 𝑥 ) = ∅ ) |
14 |
13
|
necon3bii |
⊢ ( 𝑥 ≠ ∅ ↔ ( rank ‘ 𝑥 ) ≠ ∅ ) |
15 |
11 14
|
sylibr |
⊢ ( ( rank ‘ 𝑥 ) = 1o → 𝑥 ≠ ∅ ) |
16 |
12
|
rankval |
⊢ ( rank ‘ 𝑥 ) = ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } |
17 |
16
|
eqeq1i |
⊢ ( ( rank ‘ 𝑥 ) = 1o ↔ ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o ) |
18 |
|
ssrab2 |
⊢ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ⊆ On |
19 |
|
elirr |
⊢ ¬ 1o ∈ 1o |
20 |
|
1oex |
⊢ 1o ∈ V |
21 |
|
id |
⊢ ( V = 1o → V = 1o ) |
22 |
20 21
|
eleqtrid |
⊢ ( V = 1o → 1o ∈ 1o ) |
23 |
19 22
|
mto |
⊢ ¬ V = 1o |
24 |
|
inteq |
⊢ ( { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = ∅ → ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = ∩ ∅ ) |
25 |
|
int0 |
⊢ ∩ ∅ = V |
26 |
24 25
|
eqtrdi |
⊢ ( { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = ∅ → ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = V ) |
27 |
26
|
eqeq1d |
⊢ ( { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = ∅ → ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o ↔ V = 1o ) ) |
28 |
23 27
|
mtbiri |
⊢ ( { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = ∅ → ¬ ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o ) |
29 |
28
|
necon2ai |
⊢ ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o → { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ≠ ∅ ) |
30 |
|
onint |
⊢ ( ( { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ⊆ On ∧ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ≠ ∅ ) → ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ∈ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ) |
31 |
18 29 30
|
sylancr |
⊢ ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o → ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ∈ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ) |
32 |
|
eleq1 |
⊢ ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o → ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ∈ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ↔ 1o ∈ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ) ) |
33 |
31 32
|
mpbid |
⊢ ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o → 1o ∈ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ) |
34 |
|
suceq |
⊢ ( 𝑦 = 1o → suc 𝑦 = suc 1o ) |
35 |
34
|
fveq2d |
⊢ ( 𝑦 = 1o → ( 𝑅1 ‘ suc 𝑦 ) = ( 𝑅1 ‘ suc 1o ) ) |
36 |
|
df-1o |
⊢ 1o = suc ∅ |
37 |
36
|
fveq2i |
⊢ ( 𝑅1 ‘ 1o ) = ( 𝑅1 ‘ suc ∅ ) |
38 |
|
0elon |
⊢ ∅ ∈ On |
39 |
|
r1suc |
⊢ ( ∅ ∈ On → ( 𝑅1 ‘ suc ∅ ) = 𝒫 ( 𝑅1 ‘ ∅ ) ) |
40 |
38 39
|
ax-mp |
⊢ ( 𝑅1 ‘ suc ∅ ) = 𝒫 ( 𝑅1 ‘ ∅ ) |
41 |
|
r10 |
⊢ ( 𝑅1 ‘ ∅ ) = ∅ |
42 |
41
|
pweqi |
⊢ 𝒫 ( 𝑅1 ‘ ∅ ) = 𝒫 ∅ |
43 |
37 40 42
|
3eqtri |
⊢ ( 𝑅1 ‘ 1o ) = 𝒫 ∅ |
44 |
43
|
pweqi |
⊢ 𝒫 ( 𝑅1 ‘ 1o ) = 𝒫 𝒫 ∅ |
45 |
|
pw0 |
⊢ 𝒫 ∅ = { ∅ } |
46 |
45
|
pweqi |
⊢ 𝒫 𝒫 ∅ = 𝒫 { ∅ } |
47 |
|
pwpw0 |
⊢ 𝒫 { ∅ } = { ∅ , { ∅ } } |
48 |
44 46 47
|
3eqtrri |
⊢ { ∅ , { ∅ } } = 𝒫 ( 𝑅1 ‘ 1o ) |
49 |
|
1on |
⊢ 1o ∈ On |
50 |
|
r1suc |
⊢ ( 1o ∈ On → ( 𝑅1 ‘ suc 1o ) = 𝒫 ( 𝑅1 ‘ 1o ) ) |
51 |
49 50
|
ax-mp |
⊢ ( 𝑅1 ‘ suc 1o ) = 𝒫 ( 𝑅1 ‘ 1o ) |
52 |
48 51
|
eqtr4i |
⊢ { ∅ , { ∅ } } = ( 𝑅1 ‘ suc 1o ) |
53 |
35 52
|
eqtr4di |
⊢ ( 𝑦 = 1o → ( 𝑅1 ‘ suc 𝑦 ) = { ∅ , { ∅ } } ) |
54 |
53
|
eleq2d |
⊢ ( 𝑦 = 1o → ( 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) ↔ 𝑥 ∈ { ∅ , { ∅ } } ) ) |
55 |
54
|
elrab |
⊢ ( 1o ∈ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } ↔ ( 1o ∈ On ∧ 𝑥 ∈ { ∅ , { ∅ } } ) ) |
56 |
33 55
|
sylib |
⊢ ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o → ( 1o ∈ On ∧ 𝑥 ∈ { ∅ , { ∅ } } ) ) |
57 |
12
|
elpr |
⊢ ( 𝑥 ∈ { ∅ , { ∅ } } ↔ ( 𝑥 = ∅ ∨ 𝑥 = { ∅ } ) ) |
58 |
|
df-ne |
⊢ ( 𝑥 ≠ ∅ ↔ ¬ 𝑥 = ∅ ) |
59 |
|
orel1 |
⊢ ( ¬ 𝑥 = ∅ → ( ( 𝑥 = ∅ ∨ 𝑥 = { ∅ } ) → 𝑥 = { ∅ } ) ) |
60 |
58 59
|
sylbi |
⊢ ( 𝑥 ≠ ∅ → ( ( 𝑥 = ∅ ∨ 𝑥 = { ∅ } ) → 𝑥 = { ∅ } ) ) |
61 |
|
df1o2 |
⊢ 1o = { ∅ } |
62 |
|
eqeq2 |
⊢ ( 𝑥 = { ∅ } → ( 1o = 𝑥 ↔ 1o = { ∅ } ) ) |
63 |
61 62
|
mpbiri |
⊢ ( 𝑥 = { ∅ } → 1o = 𝑥 ) |
64 |
63
|
eqcomd |
⊢ ( 𝑥 = { ∅ } → 𝑥 = 1o ) |
65 |
60 64
|
syl6com |
⊢ ( ( 𝑥 = ∅ ∨ 𝑥 = { ∅ } ) → ( 𝑥 ≠ ∅ → 𝑥 = 1o ) ) |
66 |
57 65
|
sylbi |
⊢ ( 𝑥 ∈ { ∅ , { ∅ } } → ( 𝑥 ≠ ∅ → 𝑥 = 1o ) ) |
67 |
66
|
adantl |
⊢ ( ( 1o ∈ On ∧ 𝑥 ∈ { ∅ , { ∅ } } ) → ( 𝑥 ≠ ∅ → 𝑥 = 1o ) ) |
68 |
56 67
|
syl |
⊢ ( ∩ { 𝑦 ∈ On ∣ 𝑥 ∈ ( 𝑅1 ‘ suc 𝑦 ) } = 1o → ( 𝑥 ≠ ∅ → 𝑥 = 1o ) ) |
69 |
17 68
|
sylbi |
⊢ ( ( rank ‘ 𝑥 ) = 1o → ( 𝑥 ≠ ∅ → 𝑥 = 1o ) ) |
70 |
15 69
|
mpd |
⊢ ( ( rank ‘ 𝑥 ) = 1o → 𝑥 = 1o ) |
71 |
9 70
|
vtoclg |
⊢ ( 𝐴 ∈ V → ( ( rank ‘ 𝐴 ) = 1o → 𝐴 = 1o ) ) |
72 |
6 71
|
mpcom |
⊢ ( ( rank ‘ 𝐴 ) = 1o → 𝐴 = 1o ) |
73 |
|
fveq2 |
⊢ ( 𝐴 = 1o → ( rank ‘ 𝐴 ) = ( rank ‘ 1o ) ) |
74 |
|
r111 |
⊢ 𝑅1 : On –1-1→ V |
75 |
|
f1dm |
⊢ ( 𝑅1 : On –1-1→ V → dom 𝑅1 = On ) |
76 |
74 75
|
ax-mp |
⊢ dom 𝑅1 = On |
77 |
49 76
|
eleqtrri |
⊢ 1o ∈ dom 𝑅1 |
78 |
|
rankonid |
⊢ ( 1o ∈ dom 𝑅1 ↔ ( rank ‘ 1o ) = 1o ) |
79 |
77 78
|
mpbi |
⊢ ( rank ‘ 1o ) = 1o |
80 |
73 79
|
eqtrdi |
⊢ ( 𝐴 = 1o → ( rank ‘ 𝐴 ) = 1o ) |
81 |
72 80
|
impbii |
⊢ ( ( rank ‘ 𝐴 ) = 1o ↔ 𝐴 = 1o ) |
82 |
61
|
eqeq2i |
⊢ ( 𝐴 = 1o ↔ 𝐴 = { ∅ } ) |
83 |
81 82
|
bitri |
⊢ ( ( rank ‘ 𝐴 ) = 1o ↔ 𝐴 = { ∅ } ) |