Metamath Proof Explorer


Theorem onint1

Description: The ordinal T_1 spaces are 1o and 2o , proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015)

Ref Expression
Assertion onint1 ( On ∩ Fre ) = { 1o , 2o }

Proof

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 ( 2o𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ { ∅ } ) )
8 p0ex { ∅ } ∈ V
9 8 prid2 { ∅ } ∈ { ∅ , { ∅ } }
10 df2o2 2o = { ∅ , { ∅ } }
11 9 10 eleqtrri { ∅ } ∈ 2o
12 elunii ( ( { ∅ } ∈ 2o ∧ 2o𝑗 ) → { ∅ } ∈ 𝑗 )
13 11 12 mpan ( 2o𝑗 → { ∅ } ∈ 𝑗 )
14 df1o2 1o = { ∅ }
15 1on 1o ∈ On
16 14 15 eqeltrri { ∅ } ∈ On
17 16 onirri ¬ { ∅ } ∈ { ∅ }
18 17 a1i ( 2o𝑗 → ¬ { ∅ } ∈ { ∅ } )
19 13 18 eldifd ( 2o𝑗 → { ∅ } ∈ ( 𝑗 ∖ { ∅ } ) )
20 19 ne0d ( 2o𝑗 → ( 𝑗 ∖ { ∅ } ) ≠ ∅ )
21 7 20 2thd ( 2o𝑗 → ( ¬ ∅ ∈ ( 𝑗 ∖ { ∅ } ) ↔ ( 𝑗 ∖ { ∅ } ) ≠ ∅ ) )
22 nbbn ( ( ¬ ∅ ∈ ( 𝑗 ∖ { ∅ } ) ↔ ( 𝑗 ∖ { ∅ } ) ≠ ∅ ) ↔ ¬ ( ∅ ∈ ( 𝑗 ∖ { ∅ } ) ↔ ( 𝑗 ∖ { ∅ } ) ≠ ∅ ) )
23 21 22 sylib ( 2o𝑗 → ¬ ( ∅ ∈ ( 𝑗 ∖ { ∅ } ) ↔ ( 𝑗 ∖ { ∅ } ) ≠ ∅ ) )
24 on0eln0 ( ( 𝑗 ∖ { ∅ } ) ∈ On → ( ∅ ∈ ( 𝑗 ∖ { ∅ } ) ↔ ( 𝑗 ∖ { ∅ } ) ≠ ∅ ) )
25 23 24 nsyl ( 2o𝑗 → ¬ ( 𝑗 ∖ { ∅ } ) ∈ On )
26 6 25 nsyli ( 𝑗 ∈ On → ( 2o𝑗 → ¬ ( 𝑗 ∖ { ∅ } ) ∈ 𝑗 ) )
27 26 imp ( ( 𝑗 ∈ On ∧ 2o𝑗 ) → ¬ ( 𝑗 ∖ { ∅ } ) ∈ 𝑗 )
28 0ex ∅ ∈ V
29 28 prid1 ∅ ∈ { ∅ , { ∅ } }
30 29 10 eleqtrri ∅ ∈ 2o
31 elunii ( ( ∅ ∈ 2o ∧ 2o𝑗 ) → ∅ ∈ 𝑗 )
32 30 31 mpan ( 2o𝑗 → ∅ ∈ 𝑗 )
33 32 adantl ( ( 𝑗 ∈ On ∧ 2o𝑗 ) → ∅ ∈ 𝑗 )
34 simpr ( ( ( 𝑗 ∈ On ∧ 2o𝑗 ) ∧ 𝑎 = ∅ ) → 𝑎 = ∅ )
35 34 sneqd ( ( ( 𝑗 ∈ On ∧ 2o𝑗 ) ∧ 𝑎 = ∅ ) → { 𝑎 } = { ∅ } )
36 35 eleq1d ( ( ( 𝑗 ∈ On ∧ 2o𝑗 ) ∧ 𝑎 = ∅ ) → ( { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) ↔ { ∅ } ∈ ( Clsd ‘ 𝑗 ) ) )
37 33 36 rspcdv ( ( 𝑗 ∈ On ∧ 2o𝑗 ) → ( ∀ 𝑎 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) → { ∅ } ∈ ( Clsd ‘ 𝑗 ) ) )
38 2 cldopn ( { ∅ } ∈ ( Clsd ‘ 𝑗 ) → ( 𝑗 ∖ { ∅ } ) ∈ 𝑗 )
39 37 38 syl6 ( ( 𝑗 ∈ On ∧ 2o𝑗 ) → ( ∀ 𝑎 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) → ( 𝑗 ∖ { ∅ } ) ∈ 𝑗 ) )
40 27 39 mtod ( ( 𝑗 ∈ On ∧ 2o𝑗 ) → ¬ ∀ 𝑎 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) )
41 40 ex ( 𝑗 ∈ On → ( 2o𝑗 → ¬ ∀ 𝑎 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) ) )
42 41 con2d ( 𝑗 ∈ On → ( ∀ 𝑎 𝑗 { 𝑎 } ∈ ( Clsd ‘ 𝑗 ) → ¬ 2o𝑗 ) )
43 4 42 syl5 ( 𝑗 ∈ On → ( 𝑗 ∈ Fre → ¬ 2o𝑗 ) )
44 2on 2o ∈ On
45 ontri1 ( ( 𝑗 ∈ On ∧ 2o ∈ On ) → ( 𝑗 ⊆ 2o ↔ ¬ 2o𝑗 ) )
46 onsssuc ( ( 𝑗 ∈ On ∧ 2o ∈ On ) → ( 𝑗 ⊆ 2o𝑗 ∈ suc 2o ) )
47 45 46 bitr3d ( ( 𝑗 ∈ On ∧ 2o ∈ On ) → ( ¬ 2o𝑗𝑗 ∈ suc 2o ) )
48 44 47 mpan2 ( 𝑗 ∈ On → ( ¬ 2o𝑗𝑗 ∈ 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 }