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Theorem onint1 33857
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 3935 . . . . 5 (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2 eqid 2824 . . . . . . . . . . 11 𝑗 = 𝑗
32ist1 21932 . . . . . . . . . 10 (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
43simprbi 500 . . . . . . . . 9 (𝑗 ∈ Fre → ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
5 onelon 6203 . . . . . . . . . . . . . . 15 ((𝑗 ∈ On ∧ ( 𝑗 ∖ {∅}) ∈ 𝑗) → ( 𝑗 ∖ {∅}) ∈ On)
65ex 416 . . . . . . . . . . . . . 14 (𝑗 ∈ On → (( 𝑗 ∖ {∅}) ∈ 𝑗 → ( 𝑗 ∖ {∅}) ∈ On))
7 neldifsnd 4710 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ {∅}))
8 p0ex 5272 . . . . . . . . . . . . . . . . . . . . . 22 {∅} ∈ V
98prid2 4684 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ {∅, {∅}}
10 df2o2 8114 . . . . . . . . . . . . . . . . . . . . 21 2o = {∅, {∅}}
119, 10eleqtrri 2915 . . . . . . . . . . . . . . . . . . . 20 {∅} ∈ 2o
12 elunii 4829 . . . . . . . . . . . . . . . . . . . 20 (({∅} ∈ 2o ∧ 2o𝑗) → {∅} ∈ 𝑗)
1311, 12mpan 689 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → {∅} ∈ 𝑗)
14 df1o2 8112 . . . . . . . . . . . . . . . . . . . . . 22 1o = {∅}
15 1on 8105 . . . . . . . . . . . . . . . . . . . . . 22 1o ∈ On
1614, 15eqeltrri 2913 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ On
1716onirri 6284 . . . . . . . . . . . . . . . . . . . 20 ¬ {∅} ∈ {∅}
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → ¬ {∅} ∈ {∅})
1913, 18eldifd 3930 . . . . . . . . . . . . . . . . . 18 (2o𝑗 → {∅} ∈ ( 𝑗 ∖ {∅}))
2019ne0d 4284 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ( 𝑗 ∖ {∅}) ≠ ∅)
217, 202thd 268 . . . . . . . . . . . . . . . 16 (2o𝑗 → (¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
22 nbbn 388 . . . . . . . . . . . . . . . 16 ((¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2321, 22sylib 221 . . . . . . . . . . . . . . 15 (2o𝑗 → ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
24 on0eln0 6233 . . . . . . . . . . . . . . 15 (( 𝑗 ∖ {∅}) ∈ On → (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2523, 24nsyl 142 . . . . . . . . . . . . . 14 (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ On)
266, 25nsyli 160 . . . . . . . . . . . . 13 (𝑗 ∈ On → (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗))
2726imp 410 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗)
28 0ex 5197 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2928prid1 4683 . . . . . . . . . . . . . . . . 17 ∅ ∈ {∅, {∅}}
3029, 10eleqtrri 2915 . . . . . . . . . . . . . . . 16 ∅ ∈ 2o
31 elunii 4829 . . . . . . . . . . . . . . . 16 ((∅ ∈ 2o ∧ 2o𝑗) → ∅ ∈ 𝑗)
3230, 31mpan 689 . . . . . . . . . . . . . . 15 (2o𝑗 → ∅ ∈ 𝑗)
3332adantl 485 . . . . . . . . . . . . . 14 ((𝑗 ∈ On ∧ 2o𝑗) → ∅ ∈ 𝑗)
34 simpr 488 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
3534sneqd 4562 . . . . . . . . . . . . . . 15 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
3635eleq1d 2900 . . . . . . . . . . . . . 14 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
3733, 36rspcdv 3601 . . . . . . . . . . . . 13 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
382cldopn 21642 . . . . . . . . . . . . 13 ({∅} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗)
3937, 38syl6 35 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗))
4027, 39mtod 201 . . . . . . . . . . 11 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
4140ex 416 . . . . . . . . . 10 (𝑗 ∈ On → (2o𝑗 → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
4241con2d 136 . . . . . . . . 9 (𝑗 ∈ On → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ¬ 2o𝑗))
434, 42syl5 34 . . . . . . . 8 (𝑗 ∈ On → (𝑗 ∈ Fre → ¬ 2o𝑗))
44 2on 8107 . . . . . . . . 9 2o ∈ On
45 ontri1 6212 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o ↔ ¬ 2o𝑗))
46 onsssuc 6265 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o𝑗 ∈ suc 2o))
4745, 46bitr3d 284 . . . . . . . . 9 ((𝑗 ∈ On ∧ 2o ∈ On) → (¬ 2o𝑗𝑗 ∈ suc 2o))
4844, 47mpan2 690 . . . . . . . 8 (𝑗 ∈ On → (¬ 2o𝑗𝑗 ∈ suc 2o))
4943, 48sylibd 242 . . . . . . 7 (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2o))
5049imp 410 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2o)
51 0ntop 21516 . . . . . . . . . 10 ¬ ∅ ∈ Top
52 t1top 21941 . . . . . . . . . 10 (∅ ∈ Fre → ∅ ∈ Top)
5351, 52mto 200 . . . . . . . . 9 ¬ ∅ ∈ Fre
54 nelneq 2940 . . . . . . . . 9 ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
5553, 54mpan2 690 . . . . . . . 8 (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
56 elsni 4567 . . . . . . . 8 (𝑗 ∈ {∅} → 𝑗 = ∅)
5755, 56nsyl 142 . . . . . . 7 (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
5857adantl 485 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
5950, 58eldifd 3930 . . . . 5 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
601, 59sylbi 220 . . . 4 (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
6160ssriv 3957 . . 3 (On ∩ Fre) ⊆ (suc 2o ∖ {∅})
62 df-suc 6184 . . . . . 6 suc 2o = (2o ∪ {2o})
6362difeq1i 4081 . . . . 5 (suc 2o ∖ {∅}) = ((2o ∪ {2o}) ∖ {∅})
64 difundir 4242 . . . . 5 ((2o ∪ {2o}) ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
6563, 64eqtri 2847 . . . 4 (suc 2o ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
66 df-pr 4553 . . . . 5 {1o, 2o} = ({1o} ∪ {2o})
67 df2o3 8113 . . . . . . . . 9 2o = {∅, 1o}
68 df-pr 4553 . . . . . . . . 9 {∅, 1o} = ({∅} ∪ {1o})
6967, 68eqtri 2847 . . . . . . . 8 2o = ({∅} ∪ {1o})
7069difeq1i 4081 . . . . . . 7 (2o ∖ {∅}) = (({∅} ∪ {1o}) ∖ {∅})
71 difundir 4242 . . . . . . 7 (({∅} ∪ {1o}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅}))
72 difid 4313 . . . . . . . . 9 ({∅} ∖ {∅}) = ∅
73 1n0 8115 . . . . . . . . . . . 12 1o ≠ ∅
74 disjsn2 4633 . . . . . . . . . . . 12 (1o ≠ ∅ → ({1o} ∩ {∅}) = ∅)
7573, 74ax-mp 5 . . . . . . . . . . 11 ({1o} ∩ {∅}) = ∅
7675difeq2i 4082 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ ∅)
77 difin 4223 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ {∅})
78 dif0 4315 . . . . . . . . . 10 ({1o} ∖ ∅) = {1o}
7976, 77, 783eqtr3i 2855 . . . . . . . . 9 ({1o} ∖ {∅}) = {1o}
8072, 79uneq12i 4123 . . . . . . . 8 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = (∅ ∪ {1o})
81 uncom 4115 . . . . . . . 8 (∅ ∪ {1o}) = ({1o} ∪ ∅)
82 un0 4327 . . . . . . . 8 ({1o} ∪ ∅) = {1o}
8380, 81, 823eqtri 2851 . . . . . . 7 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = {1o}
8470, 71, 833eqtri 2851 . . . . . 6 (2o ∖ {∅}) = {1o}
85 2on0 8109 . . . . . . . . 9 2o ≠ ∅
86 disjsn2 4633 . . . . . . . . 9 (2o ≠ ∅ → ({2o} ∩ {∅}) = ∅)
8785, 86ax-mp 5 . . . . . . . 8 ({2o} ∩ {∅}) = ∅
8887difeq2i 4082 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ ∅)
89 difin 4223 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ {∅})
90 dif0 4315 . . . . . . 7 ({2o} ∖ ∅) = {2o}
9188, 89, 903eqtr3i 2855 . . . . . 6 ({2o} ∖ {∅}) = {2o}
9284, 91uneq12i 4123 . . . . 5 ((2o ∖ {∅}) ∪ ({2o} ∖ {∅})) = ({1o} ∪ {2o})
9366, 92eqtr4i 2850 . . . 4 {1o, 2o} = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
9465, 93eqtr4i 2850 . . 3 (suc 2o ∖ {∅}) = {1o, 2o}
9561, 94sseqtri 3989 . 2 (On ∩ Fre) ⊆ {1o, 2o}
96 ssoninhaus 33856 . . 3 {1o, 2o} ⊆ (On ∩ Haus)
97 haust1 21963 . . . . 5 (𝑗 ∈ Haus → 𝑗 ∈ Fre)
9897ssriv 3957 . . . 4 Haus ⊆ Fre
99 sslin 4196 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
10098, 99ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
10196, 100sstri 3962 . 2 {1o, 2o} ⊆ (On ∩ Fre)
10295, 101eqssi 3969 1 (On ∩ Fre) = {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  cdif 3916  cun 3917  cin 3918  wss 3919  c0 4276  {csn 4550  {cpr 4552   cuni 4824  Oncon0 6178  suc csuc 6180  cfv 6343  1oc1o 8091  2oc2o 8092  Topctop 21504  Clsdccld 21627  Frect1 21918  Hauscha 21919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-ord 6181  df-on 6182  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-1o 8098  df-2o 8099  df-topgen 16717  df-top 21505  df-topon 21522  df-cld 21630  df-t1 21925  df-haus 21926
This theorem is referenced by:  oninhaus  33858
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