Step |
Hyp |
Ref |
Expression |
1 |
|
elin |
⊢ ( 𝑗 ∈ ( On ∩ Fre ) ↔ ( 𝑗 ∈ On ∧ 𝑗 ∈ Fre ) ) |
2 |
|
eqid |
⊢ ∪ 𝑗 = ∪ 𝑗 |
3 |
2
|
ist1 |
⊢ ( 𝑗 ∈ Fre ↔ ( 𝑗 ∈ Top ∧ ∀ 𝑎 ∈ ∪ 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) ) ) |
4 |
3
|
simprbi |
⊢ ( 𝑗 ∈ Fre → ∀ 𝑎 ∈ ∪ 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) ) |
5 |
|
onelon |
⊢ ( ( 𝑗 ∈ On ∧ ( ∪ 𝑗 ∖ { ∅ } ) ∈ 𝑗 ) → ( ∪ 𝑗 ∖ { ∅ } ) ∈ On ) |
6 |
5
|
ex |
⊢ ( 𝑗 ∈ On → ( ( ∪ 𝑗 ∖ { ∅ } ) ∈ 𝑗 → ( ∪ 𝑗 ∖ { ∅ } ) ∈ On ) ) |
7 |
|
neldifsnd |
⊢ ( 2o ∈ 𝑗 → ¬ ∅ ∈ ( ∪ 𝑗 ∖ { ∅ } ) ) |
8 |
|
p0ex |
⊢ { ∅ } ∈ V |
9 |
8
|
prid2 |
⊢ { ∅ } ∈ { ∅ , { ∅ } } |
10 |
|
df2o2 |
⊢ 2o = { ∅ , { ∅ } } |
11 |
9 10
|
eleqtrri |
⊢ { ∅ } ∈ 2o |
12 |
|
elunii |
⊢ ( ( { ∅ } ∈ 2o ∧ 2o ∈ 𝑗 ) → { ∅ } ∈ ∪ 𝑗 ) |
13 |
11 12
|
mpan |
⊢ ( 2o ∈ 𝑗 → { ∅ } ∈ ∪ 𝑗 ) |
14 |
|
df1o2 |
⊢ 1o = { ∅ } |
15 |
|
1on |
⊢ 1o ∈ On |
16 |
14 15
|
eqeltrri |
⊢ { ∅ } ∈ On |
17 |
16
|
onirri |
⊢ ¬ { ∅ } ∈ { ∅ } |
18 |
17
|
a1i |
⊢ ( 2o ∈ 𝑗 → ¬ { ∅ } ∈ { ∅ } ) |
19 |
13 18
|
eldifd |
⊢ ( 2o ∈ 𝑗 → { ∅ } ∈ ( ∪ 𝑗 ∖ { ∅ } ) ) |
20 |
19
|
ne0d |
⊢ ( 2o ∈ 𝑗 → ( ∪ 𝑗 ∖ { ∅ } ) ≠ ∅ ) |
21 |
7 20
|
2thd |
⊢ ( 2o ∈ 𝑗 → ( ¬ ∅ ∈ ( ∪ 𝑗 ∖ { ∅ } ) ↔ ( ∪ 𝑗 ∖ { ∅ } ) ≠ ∅ ) ) |
22 |
|
nbbn |
⊢ ( ( ¬ ∅ ∈ ( ∪ 𝑗 ∖ { ∅ } ) ↔ ( ∪ 𝑗 ∖ { ∅ } ) ≠ ∅ ) ↔ ¬ ( ∅ ∈ ( ∪ 𝑗 ∖ { ∅ } ) ↔ ( ∪ 𝑗 ∖ { ∅ } ) ≠ ∅ ) ) |
23 |
21 22
|
sylib |
⊢ ( 2o ∈ 𝑗 → ¬ ( ∅ ∈ ( ∪ 𝑗 ∖ { ∅ } ) ↔ ( ∪ 𝑗 ∖ { ∅ } ) ≠ ∅ ) ) |
24 |
|
on0eln0 |
⊢ ( ( ∪ 𝑗 ∖ { ∅ } ) ∈ On → ( ∅ ∈ ( ∪ 𝑗 ∖ { ∅ } ) ↔ ( ∪ 𝑗 ∖ { ∅ } ) ≠ ∅ ) ) |
25 |
23 24
|
nsyl |
⊢ ( 2o ∈ 𝑗 → ¬ ( ∪ 𝑗 ∖ { ∅ } ) ∈ On ) |
26 |
6 25
|
nsyli |
⊢ ( 𝑗 ∈ On → ( 2o ∈ 𝑗 → ¬ ( ∪ 𝑗 ∖ { ∅ } ) ∈ 𝑗 ) ) |
27 |
26
|
imp |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) → ¬ ( ∪ 𝑗 ∖ { ∅ } ) ∈ 𝑗 ) |
28 |
|
0ex |
⊢ ∅ ∈ V |
29 |
28
|
prid1 |
⊢ ∅ ∈ { ∅ , { ∅ } } |
30 |
29 10
|
eleqtrri |
⊢ ∅ ∈ 2o |
31 |
|
elunii |
⊢ ( ( ∅ ∈ 2o ∧ 2o ∈ 𝑗 ) → ∅ ∈ ∪ 𝑗 ) |
32 |
30 31
|
mpan |
⊢ ( 2o ∈ 𝑗 → ∅ ∈ ∪ 𝑗 ) |
33 |
32
|
adantl |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) → ∅ ∈ ∪ 𝑗 ) |
34 |
|
simpr |
⊢ ( ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) ∧ 𝑎 = ∅ ) → 𝑎 = ∅ ) |
35 |
34
|
sneqd |
⊢ ( ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) ∧ 𝑎 = ∅ ) → { 𝑎 } = { ∅ } ) |
36 |
35
|
eleq1d |
⊢ ( ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) ∧ 𝑎 = ∅ ) → ( { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) ↔ { ∅ } ∈ ( Clsd ‘ 𝑗 ) ) ) |
37 |
33 36
|
rspcdv |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) → ( ∀ 𝑎 ∈ ∪ 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) → { ∅ } ∈ ( Clsd ‘ 𝑗 ) ) ) |
38 |
2
|
cldopn |
⊢ ( { ∅ } ∈ ( Clsd ‘ 𝑗 ) → ( ∪ 𝑗 ∖ { ∅ } ) ∈ 𝑗 ) |
39 |
37 38
|
syl6 |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) → ( ∀ 𝑎 ∈ ∪ 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) → ( ∪ 𝑗 ∖ { ∅ } ) ∈ 𝑗 ) ) |
40 |
27 39
|
mtod |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ 𝑗 ) → ¬ ∀ 𝑎 ∈ ∪ 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) ) |
41 |
40
|
ex |
⊢ ( 𝑗 ∈ On → ( 2o ∈ 𝑗 → ¬ ∀ 𝑎 ∈ ∪ 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) ) ) |
42 |
41
|
con2d |
⊢ ( 𝑗 ∈ On → ( ∀ 𝑎 ∈ ∪ 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) → ¬ 2o ∈ 𝑗 ) ) |
43 |
4 42
|
syl5 |
⊢ ( 𝑗 ∈ On → ( 𝑗 ∈ Fre → ¬ 2o ∈ 𝑗 ) ) |
44 |
|
2on |
⊢ 2o ∈ On |
45 |
|
ontri1 |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ On ) → ( 𝑗 ⊆ 2o ↔ ¬ 2o ∈ 𝑗 ) ) |
46 |
|
onsssuc |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ On ) → ( 𝑗 ⊆ 2o ↔ 𝑗 ∈ suc 2o ) ) |
47 |
45 46
|
bitr3d |
⊢ ( ( 𝑗 ∈ On ∧ 2o ∈ On ) → ( ¬ 2o ∈ 𝑗 ↔ 𝑗 ∈ suc 2o ) ) |
48 |
44 47
|
mpan2 |
⊢ ( 𝑗 ∈ On → ( ¬ 2o ∈ 𝑗 ↔ 𝑗 ∈ suc 2o ) ) |
49 |
43 48
|
sylibd |
⊢ ( 𝑗 ∈ On → ( 𝑗 ∈ Fre → 𝑗 ∈ suc 2o ) ) |
50 |
49
|
imp |
⊢ ( ( 𝑗 ∈ On ∧ 𝑗 ∈ Fre ) → 𝑗 ∈ suc 2o ) |
51 |
|
0ntop |
⊢ ¬ ∅ ∈ Top |
52 |
|
t1top |
⊢ ( ∅ ∈ Fre → ∅ ∈ Top ) |
53 |
51 52
|
mto |
⊢ ¬ ∅ ∈ Fre |
54 |
|
nelneq |
⊢ ( ( 𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre ) → ¬ 𝑗 = ∅ ) |
55 |
53 54
|
mpan2 |
⊢ ( 𝑗 ∈ Fre → ¬ 𝑗 = ∅ ) |
56 |
|
elsni |
⊢ ( 𝑗 ∈ { ∅ } → 𝑗 = ∅ ) |
57 |
55 56
|
nsyl |
⊢ ( 𝑗 ∈ Fre → ¬ 𝑗 ∈ { ∅ } ) |
58 |
57
|
adantl |
⊢ ( ( 𝑗 ∈ On ∧ 𝑗 ∈ Fre ) → ¬ 𝑗 ∈ { ∅ } ) |
59 |
50 58
|
eldifd |
⊢ ( ( 𝑗 ∈ On ∧ 𝑗 ∈ Fre ) → 𝑗 ∈ ( suc 2o ∖ { ∅ } ) ) |
60 |
1 59
|
sylbi |
⊢ ( 𝑗 ∈ ( On ∩ Fre ) → 𝑗 ∈ ( suc 2o ∖ { ∅ } ) ) |
61 |
60
|
ssriv |
⊢ ( On ∩ Fre ) ⊆ ( suc 2o ∖ { ∅ } ) |
62 |
|
df-suc |
⊢ suc 2o = ( 2o ∪ { 2o } ) |
63 |
62
|
difeq1i |
⊢ ( suc 2o ∖ { ∅ } ) = ( ( 2o ∪ { 2o } ) ∖ { ∅ } ) |
64 |
|
difundir |
⊢ ( ( 2o ∪ { 2o } ) ∖ { ∅ } ) = ( ( 2o ∖ { ∅ } ) ∪ ( { 2o } ∖ { ∅ } ) ) |
65 |
63 64
|
eqtri |
⊢ ( suc 2o ∖ { ∅ } ) = ( ( 2o ∖ { ∅ } ) ∪ ( { 2o } ∖ { ∅ } ) ) |
66 |
|
df-pr |
⊢ { 1o , 2o } = ( { 1o } ∪ { 2o } ) |
67 |
|
df2o3 |
⊢ 2o = { ∅ , 1o } |
68 |
|
df-pr |
⊢ { ∅ , 1o } = ( { ∅ } ∪ { 1o } ) |
69 |
67 68
|
eqtri |
⊢ 2o = ( { ∅ } ∪ { 1o } ) |
70 |
69
|
difeq1i |
⊢ ( 2o ∖ { ∅ } ) = ( ( { ∅ } ∪ { 1o } ) ∖ { ∅ } ) |
71 |
|
difundir |
⊢ ( ( { ∅ } ∪ { 1o } ) ∖ { ∅ } ) = ( ( { ∅ } ∖ { ∅ } ) ∪ ( { 1o } ∖ { ∅ } ) ) |
72 |
|
difid |
⊢ ( { ∅ } ∖ { ∅ } ) = ∅ |
73 |
|
1n0 |
⊢ 1o ≠ ∅ |
74 |
|
disjsn2 |
⊢ ( 1o ≠ ∅ → ( { 1o } ∩ { ∅ } ) = ∅ ) |
75 |
73 74
|
ax-mp |
⊢ ( { 1o } ∩ { ∅ } ) = ∅ |
76 |
75
|
difeq2i |
⊢ ( { 1o } ∖ ( { 1o } ∩ { ∅ } ) ) = ( { 1o } ∖ ∅ ) |
77 |
|
difin |
⊢ ( { 1o } ∖ ( { 1o } ∩ { ∅ } ) ) = ( { 1o } ∖ { ∅ } ) |
78 |
|
dif0 |
⊢ ( { 1o } ∖ ∅ ) = { 1o } |
79 |
76 77 78
|
3eqtr3i |
⊢ ( { 1o } ∖ { ∅ } ) = { 1o } |
80 |
72 79
|
uneq12i |
⊢ ( ( { ∅ } ∖ { ∅ } ) ∪ ( { 1o } ∖ { ∅ } ) ) = ( ∅ ∪ { 1o } ) |
81 |
|
uncom |
⊢ ( ∅ ∪ { 1o } ) = ( { 1o } ∪ ∅ ) |
82 |
|
un0 |
⊢ ( { 1o } ∪ ∅ ) = { 1o } |
83 |
80 81 82
|
3eqtri |
⊢ ( ( { ∅ } ∖ { ∅ } ) ∪ ( { 1o } ∖ { ∅ } ) ) = { 1o } |
84 |
70 71 83
|
3eqtri |
⊢ ( 2o ∖ { ∅ } ) = { 1o } |
85 |
|
2on0 |
⊢ 2o ≠ ∅ |
86 |
|
disjsn2 |
⊢ ( 2o ≠ ∅ → ( { 2o } ∩ { ∅ } ) = ∅ ) |
87 |
85 86
|
ax-mp |
⊢ ( { 2o } ∩ { ∅ } ) = ∅ |
88 |
87
|
difeq2i |
⊢ ( { 2o } ∖ ( { 2o } ∩ { ∅ } ) ) = ( { 2o } ∖ ∅ ) |
89 |
|
difin |
⊢ ( { 2o } ∖ ( { 2o } ∩ { ∅ } ) ) = ( { 2o } ∖ { ∅ } ) |
90 |
|
dif0 |
⊢ ( { 2o } ∖ ∅ ) = { 2o } |
91 |
88 89 90
|
3eqtr3i |
⊢ ( { 2o } ∖ { ∅ } ) = { 2o } |
92 |
84 91
|
uneq12i |
⊢ ( ( 2o ∖ { ∅ } ) ∪ ( { 2o } ∖ { ∅ } ) ) = ( { 1o } ∪ { 2o } ) |
93 |
66 92
|
eqtr4i |
⊢ { 1o , 2o } = ( ( 2o ∖ { ∅ } ) ∪ ( { 2o } ∖ { ∅ } ) ) |
94 |
65 93
|
eqtr4i |
⊢ ( suc 2o ∖ { ∅ } ) = { 1o , 2o } |
95 |
61 94
|
sseqtri |
⊢ ( On ∩ Fre ) ⊆ { 1o , 2o } |
96 |
|
ssoninhaus |
⊢ { 1o , 2o } ⊆ ( On ∩ Haus ) |
97 |
|
haust1 |
⊢ ( 𝑗 ∈ Haus → 𝑗 ∈ Fre ) |
98 |
97
|
ssriv |
⊢ Haus ⊆ Fre |
99 |
|
sslin |
⊢ ( Haus ⊆ Fre → ( On ∩ Haus ) ⊆ ( On ∩ Fre ) ) |
100 |
98 99
|
ax-mp |
⊢ ( On ∩ Haus ) ⊆ ( On ∩ Fre ) |
101 |
96 100
|
sstri |
⊢ { 1o , 2o } ⊆ ( On ∩ Fre ) |
102 |
95 101
|
eqssi |
⊢ ( On ∩ Fre ) = { 1o , 2o } |