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Theorem onint1 33317
Description: The ordinal T1 spaces are 1o and 2o, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)
Assertion
Ref Expression
onint1 (On ∩ Fre) = {1o, 2o}

Proof of Theorem onint1
Dummy variables 𝑗 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 4051 . . . . 5 (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2 eqid 2772 . . . . . . . . . . 11 𝑗 = 𝑗
32ist1 21627 . . . . . . . . . 10 (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
43simprbi 489 . . . . . . . . 9 (𝑗 ∈ Fre → ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
5 onelon 6048 . . . . . . . . . . . . . . 15 ((𝑗 ∈ On ∧ ( 𝑗 ∖ {∅}) ∈ 𝑗) → ( 𝑗 ∖ {∅}) ∈ On)
65ex 405 . . . . . . . . . . . . . 14 (𝑗 ∈ On → (( 𝑗 ∖ {∅}) ∈ 𝑗 → ( 𝑗 ∖ {∅}) ∈ On))
7 neldifsnd 4594 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ {∅}))
8 p0ex 5131 . . . . . . . . . . . . . . . . . . . . . 22 {∅} ∈ V
98prid2 4567 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ {∅, {∅}}
10 df2o2 7914 . . . . . . . . . . . . . . . . . . . . 21 2o = {∅, {∅}}
119, 10eleqtrri 2859 . . . . . . . . . . . . . . . . . . . 20 {∅} ∈ 2o
12 elunii 4711 . . . . . . . . . . . . . . . . . . . 20 (({∅} ∈ 2o ∧ 2o𝑗) → {∅} ∈ 𝑗)
1311, 12mpan 677 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → {∅} ∈ 𝑗)
14 df1o2 7912 . . . . . . . . . . . . . . . . . . . . . 22 1o = {∅}
15 1on 7906 . . . . . . . . . . . . . . . . . . . . . 22 1o ∈ On
1614, 15eqeltrri 2857 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ On
1716onirri 6129 . . . . . . . . . . . . . . . . . . . 20 ¬ {∅} ∈ {∅}
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → ¬ {∅} ∈ {∅})
1913, 18eldifd 3834 . . . . . . . . . . . . . . . . . 18 (2o𝑗 → {∅} ∈ ( 𝑗 ∖ {∅}))
2019ne0d 4181 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ( 𝑗 ∖ {∅}) ≠ ∅)
217, 202thd 257 . . . . . . . . . . . . . . . 16 (2o𝑗 → (¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
22 nbbn 376 . . . . . . . . . . . . . . . 16 ((¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2321, 22sylib 210 . . . . . . . . . . . . . . 15 (2o𝑗 → ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
24 on0eln0 6078 . . . . . . . . . . . . . . 15 (( 𝑗 ∖ {∅}) ∈ On → (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2523, 24nsyl 138 . . . . . . . . . . . . . 14 (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ On)
266, 25nsyli 157 . . . . . . . . . . . . 13 (𝑗 ∈ On → (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗))
2726imp 398 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗)
28 0ex 5062 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2928prid1 4566 . . . . . . . . . . . . . . . . 17 ∅ ∈ {∅, {∅}}
3029, 10eleqtrri 2859 . . . . . . . . . . . . . . . 16 ∅ ∈ 2o
31 elunii 4711 . . . . . . . . . . . . . . . 16 ((∅ ∈ 2o ∧ 2o𝑗) → ∅ ∈ 𝑗)
3230, 31mpan 677 . . . . . . . . . . . . . . 15 (2o𝑗 → ∅ ∈ 𝑗)
3332adantl 474 . . . . . . . . . . . . . 14 ((𝑗 ∈ On ∧ 2o𝑗) → ∅ ∈ 𝑗)
34 simpr 477 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
3534sneqd 4447 . . . . . . . . . . . . . . 15 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
3635eleq1d 2844 . . . . . . . . . . . . . 14 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
3733, 36rspcdv 3532 . . . . . . . . . . . . 13 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
382cldopn 21337 . . . . . . . . . . . . 13 ({∅} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗)
3937, 38syl6 35 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗))
4027, 39mtod 190 . . . . . . . . . . 11 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
4140ex 405 . . . . . . . . . 10 (𝑗 ∈ On → (2o𝑗 → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
4241con2d 132 . . . . . . . . 9 (𝑗 ∈ On → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ¬ 2o𝑗))
434, 42syl5 34 . . . . . . . 8 (𝑗 ∈ On → (𝑗 ∈ Fre → ¬ 2o𝑗))
44 2on 7908 . . . . . . . . 9 2o ∈ On
45 ontri1 6057 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o ↔ ¬ 2o𝑗))
46 onsssuc 6110 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o𝑗 ∈ suc 2o))
4745, 46bitr3d 273 . . . . . . . . 9 ((𝑗 ∈ On ∧ 2o ∈ On) → (¬ 2o𝑗𝑗 ∈ suc 2o))
4844, 47mpan2 678 . . . . . . . 8 (𝑗 ∈ On → (¬ 2o𝑗𝑗 ∈ suc 2o))
4943, 48sylibd 231 . . . . . . 7 (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2o))
5049imp 398 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2o)
51 0ntop 21211 . . . . . . . . . 10 ¬ ∅ ∈ Top
52 t1top 21636 . . . . . . . . . 10 (∅ ∈ Fre → ∅ ∈ Top)
5351, 52mto 189 . . . . . . . . 9 ¬ ∅ ∈ Fre
54 nelneq 2884 . . . . . . . . 9 ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
5553, 54mpan2 678 . . . . . . . 8 (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
56 elsni 4452 . . . . . . . 8 (𝑗 ∈ {∅} → 𝑗 = ∅)
5755, 56nsyl 138 . . . . . . 7 (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
5857adantl 474 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
5950, 58eldifd 3834 . . . . 5 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
601, 59sylbi 209 . . . 4 (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
6160ssriv 3856 . . 3 (On ∩ Fre) ⊆ (suc 2o ∖ {∅})
62 df-suc 6029 . . . . . 6 suc 2o = (2o ∪ {2o})
6362difeq1i 3979 . . . . 5 (suc 2o ∖ {∅}) = ((2o ∪ {2o}) ∖ {∅})
64 difundir 4138 . . . . 5 ((2o ∪ {2o}) ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
6563, 64eqtri 2796 . . . 4 (suc 2o ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
66 df-pr 4438 . . . . 5 {1o, 2o} = ({1o} ∪ {2o})
67 df2o3 7913 . . . . . . . . 9 2o = {∅, 1o}
68 df-pr 4438 . . . . . . . . 9 {∅, 1o} = ({∅} ∪ {1o})
6967, 68eqtri 2796 . . . . . . . 8 2o = ({∅} ∪ {1o})
7069difeq1i 3979 . . . . . . 7 (2o ∖ {∅}) = (({∅} ∪ {1o}) ∖ {∅})
71 difundir 4138 . . . . . . 7 (({∅} ∪ {1o}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅}))
72 difid 4210 . . . . . . . . 9 ({∅} ∖ {∅}) = ∅
73 1n0 7915 . . . . . . . . . . . 12 1o ≠ ∅
74 disjsn2 4516 . . . . . . . . . . . 12 (1o ≠ ∅ → ({1o} ∩ {∅}) = ∅)
7573, 74ax-mp 5 . . . . . . . . . . 11 ({1o} ∩ {∅}) = ∅
7675difeq2i 3980 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ ∅)
77 difin 4119 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ {∅})
78 dif0 4212 . . . . . . . . . 10 ({1o} ∖ ∅) = {1o}
7976, 77, 783eqtr3i 2804 . . . . . . . . 9 ({1o} ∖ {∅}) = {1o}
8072, 79uneq12i 4020 . . . . . . . 8 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = (∅ ∪ {1o})
81 uncom 4012 . . . . . . . 8 (∅ ∪ {1o}) = ({1o} ∪ ∅)
82 un0 4224 . . . . . . . 8 ({1o} ∪ ∅) = {1o}
8380, 81, 823eqtri 2800 . . . . . . 7 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = {1o}
8470, 71, 833eqtri 2800 . . . . . 6 (2o ∖ {∅}) = {1o}
85 2on0 7909 . . . . . . . . 9 2o ≠ ∅
86 disjsn2 4516 . . . . . . . . 9 (2o ≠ ∅ → ({2o} ∩ {∅}) = ∅)
8785, 86ax-mp 5 . . . . . . . 8 ({2o} ∩ {∅}) = ∅
8887difeq2i 3980 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ ∅)
89 difin 4119 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ {∅})
90 dif0 4212 . . . . . . 7 ({2o} ∖ ∅) = {2o}
9188, 89, 903eqtr3i 2804 . . . . . 6 ({2o} ∖ {∅}) = {2o}
9284, 91uneq12i 4020 . . . . 5 ((2o ∖ {∅}) ∪ ({2o} ∖ {∅})) = ({1o} ∪ {2o})
9366, 92eqtr4i 2799 . . . 4 {1o, 2o} = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
9465, 93eqtr4i 2799 . . 3 (suc 2o ∖ {∅}) = {1o, 2o}
9561, 94sseqtri 3887 . 2 (On ∩ Fre) ⊆ {1o, 2o}
96 ssoninhaus 33316 . . 3 {1o, 2o} ⊆ (On ∩ Haus)
97 haust1 21658 . . . . 5 (𝑗 ∈ Haus → 𝑗 ∈ Fre)
9897ssriv 3856 . . . 4 Haus ⊆ Fre
99 sslin 4092 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
10098, 99ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
10196, 100sstri 3861 . 2 {1o, 2o} ⊆ (On ∩ Fre)
10295, 101eqssi 3868 1 (On ∩ Fre) = {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 387   = wceq 1507  wcel 2050  wne 2961  wral 3082  cdif 3820  cun 3821  cin 3822  wss 3823  c0 4172  {csn 4435  {cpr 4437   cuni 4706  Oncon0 6023  suc csuc 6025  cfv 6182  1oc1o 7892  2oc2o 7893  Topctop 21199  Clsdccld 21322  Frect1 21613  Hauscha 21614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-ord 6026  df-on 6027  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-fv 6190  df-1o 7899  df-2o 7900  df-topgen 16567  df-top 21200  df-topon 21217  df-cld 21325  df-t1 21620  df-haus 21621
This theorem is referenced by:  oninhaus  33318
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