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Theorem onint1 36431
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 3978 . . . . 5 (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2 eqid 2734 . . . . . . . . . . 11 𝑗 = 𝑗
32ist1 23344 . . . . . . . . . 10 (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
43simprbi 496 . . . . . . . . 9 (𝑗 ∈ Fre → ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
5 onelon 6410 . . . . . . . . . . . . . . 15 ((𝑗 ∈ On ∧ ( 𝑗 ∖ {∅}) ∈ 𝑗) → ( 𝑗 ∖ {∅}) ∈ On)
65ex 412 . . . . . . . . . . . . . 14 (𝑗 ∈ On → (( 𝑗 ∖ {∅}) ∈ 𝑗 → ( 𝑗 ∖ {∅}) ∈ On))
7 neldifsnd 4797 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ {∅}))
8 p0ex 5389 . . . . . . . . . . . . . . . . . . . . . 22 {∅} ∈ V
98prid2 4767 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ {∅, {∅}}
10 df2o2 8513 . . . . . . . . . . . . . . . . . . . . 21 2o = {∅, {∅}}
119, 10eleqtrri 2837 . . . . . . . . . . . . . . . . . . . 20 {∅} ∈ 2o
12 elunii 4916 . . . . . . . . . . . . . . . . . . . 20 (({∅} ∈ 2o ∧ 2o𝑗) → {∅} ∈ 𝑗)
1311, 12mpan 690 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → {∅} ∈ 𝑗)
14 df1o2 8511 . . . . . . . . . . . . . . . . . . . . . 22 1o = {∅}
15 1on 8516 . . . . . . . . . . . . . . . . . . . . . 22 1o ∈ On
1614, 15eqeltrri 2835 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ On
1716onirri 6498 . . . . . . . . . . . . . . . . . . . 20 ¬ {∅} ∈ {∅}
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → ¬ {∅} ∈ {∅})
1913, 18eldifd 3973 . . . . . . . . . . . . . . . . . 18 (2o𝑗 → {∅} ∈ ( 𝑗 ∖ {∅}))
2019ne0d 4347 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ( 𝑗 ∖ {∅}) ≠ ∅)
217, 202thd 265 . . . . . . . . . . . . . . . 16 (2o𝑗 → (¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
22 nbbn 383 . . . . . . . . . . . . . . . 16 ((¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2321, 22sylib 218 . . . . . . . . . . . . . . 15 (2o𝑗 → ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
24 on0eln0 6441 . . . . . . . . . . . . . . 15 (( 𝑗 ∖ {∅}) ∈ On → (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2523, 24nsyl 140 . . . . . . . . . . . . . 14 (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ On)
266, 25nsyli 157 . . . . . . . . . . . . 13 (𝑗 ∈ On → (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗))
2726imp 406 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗)
28 0ex 5312 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2928prid1 4766 . . . . . . . . . . . . . . . . 17 ∅ ∈ {∅, {∅}}
3029, 10eleqtrri 2837 . . . . . . . . . . . . . . . 16 ∅ ∈ 2o
31 elunii 4916 . . . . . . . . . . . . . . . 16 ((∅ ∈ 2o ∧ 2o𝑗) → ∅ ∈ 𝑗)
3230, 31mpan 690 . . . . . . . . . . . . . . 15 (2o𝑗 → ∅ ∈ 𝑗)
3332adantl 481 . . . . . . . . . . . . . 14 ((𝑗 ∈ On ∧ 2o𝑗) → ∅ ∈ 𝑗)
34 simpr 484 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
3534sneqd 4642 . . . . . . . . . . . . . . 15 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
3635eleq1d 2823 . . . . . . . . . . . . . 14 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
3733, 36rspcdv 3613 . . . . . . . . . . . . 13 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
382cldopn 23054 . . . . . . . . . . . . 13 ({∅} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗)
3937, 38syl6 35 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗))
4027, 39mtod 198 . . . . . . . . . . 11 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
4140ex 412 . . . . . . . . . 10 (𝑗 ∈ On → (2o𝑗 → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
4241con2d 134 . . . . . . . . 9 (𝑗 ∈ On → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ¬ 2o𝑗))
434, 42syl5 34 . . . . . . . 8 (𝑗 ∈ On → (𝑗 ∈ Fre → ¬ 2o𝑗))
44 2on 8518 . . . . . . . . 9 2o ∈ On
45 ontri1 6419 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o ↔ ¬ 2o𝑗))
46 onsssuc 6475 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o𝑗 ∈ suc 2o))
4745, 46bitr3d 281 . . . . . . . . 9 ((𝑗 ∈ On ∧ 2o ∈ On) → (¬ 2o𝑗𝑗 ∈ suc 2o))
4844, 47mpan2 691 . . . . . . . 8 (𝑗 ∈ On → (¬ 2o𝑗𝑗 ∈ suc 2o))
4943, 48sylibd 239 . . . . . . 7 (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2o))
5049imp 406 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2o)
51 0ntop 22926 . . . . . . . . . 10 ¬ ∅ ∈ Top
52 t1top 23353 . . . . . . . . . 10 (∅ ∈ Fre → ∅ ∈ Top)
5351, 52mto 197 . . . . . . . . 9 ¬ ∅ ∈ Fre
54 nelneq 2862 . . . . . . . . 9 ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
5553, 54mpan2 691 . . . . . . . 8 (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
56 elsni 4647 . . . . . . . 8 (𝑗 ∈ {∅} → 𝑗 = ∅)
5755, 56nsyl 140 . . . . . . 7 (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
5857adantl 481 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
5950, 58eldifd 3973 . . . . 5 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
601, 59sylbi 217 . . . 4 (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
6160ssriv 3998 . . 3 (On ∩ Fre) ⊆ (suc 2o ∖ {∅})
62 df-suc 6391 . . . . . 6 suc 2o = (2o ∪ {2o})
6362difeq1i 4131 . . . . 5 (suc 2o ∖ {∅}) = ((2o ∪ {2o}) ∖ {∅})
64 difundir 4296 . . . . 5 ((2o ∪ {2o}) ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
6563, 64eqtri 2762 . . . 4 (suc 2o ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
66 df-pr 4633 . . . . 5 {1o, 2o} = ({1o} ∪ {2o})
67 df2o3 8512 . . . . . . . . 9 2o = {∅, 1o}
68 df-pr 4633 . . . . . . . . 9 {∅, 1o} = ({∅} ∪ {1o})
6967, 68eqtri 2762 . . . . . . . 8 2o = ({∅} ∪ {1o})
7069difeq1i 4131 . . . . . . 7 (2o ∖ {∅}) = (({∅} ∪ {1o}) ∖ {∅})
71 difundir 4296 . . . . . . 7 (({∅} ∪ {1o}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅}))
72 difid 4381 . . . . . . . . 9 ({∅} ∖ {∅}) = ∅
73 1n0 8524 . . . . . . . . . . . 12 1o ≠ ∅
74 disjsn2 4716 . . . . . . . . . . . 12 (1o ≠ ∅ → ({1o} ∩ {∅}) = ∅)
7573, 74ax-mp 5 . . . . . . . . . . 11 ({1o} ∩ {∅}) = ∅
7675difeq2i 4132 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ ∅)
77 difin 4277 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ {∅})
78 dif0 4383 . . . . . . . . . 10 ({1o} ∖ ∅) = {1o}
7976, 77, 783eqtr3i 2770 . . . . . . . . 9 ({1o} ∖ {∅}) = {1o}
8072, 79uneq12i 4175 . . . . . . . 8 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = (∅ ∪ {1o})
81 uncom 4167 . . . . . . . 8 (∅ ∪ {1o}) = ({1o} ∪ ∅)
82 un0 4399 . . . . . . . 8 ({1o} ∪ ∅) = {1o}
8380, 81, 823eqtri 2766 . . . . . . 7 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = {1o}
8470, 71, 833eqtri 2766 . . . . . 6 (2o ∖ {∅}) = {1o}
85 2on0 8520 . . . . . . . . 9 2o ≠ ∅
86 disjsn2 4716 . . . . . . . . 9 (2o ≠ ∅ → ({2o} ∩ {∅}) = ∅)
8785, 86ax-mp 5 . . . . . . . 8 ({2o} ∩ {∅}) = ∅
8887difeq2i 4132 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ ∅)
89 difin 4277 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ {∅})
90 dif0 4383 . . . . . . 7 ({2o} ∖ ∅) = {2o}
9188, 89, 903eqtr3i 2770 . . . . . 6 ({2o} ∖ {∅}) = {2o}
9284, 91uneq12i 4175 . . . . 5 ((2o ∖ {∅}) ∪ ({2o} ∖ {∅})) = ({1o} ∪ {2o})
9366, 92eqtr4i 2765 . . . 4 {1o, 2o} = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
9465, 93eqtr4i 2765 . . 3 (suc 2o ∖ {∅}) = {1o, 2o}
9561, 94sseqtri 4031 . 2 (On ∩ Fre) ⊆ {1o, 2o}
96 ssoninhaus 36430 . . 3 {1o, 2o} ⊆ (On ∩ Haus)
97 haust1 23375 . . . . 5 (𝑗 ∈ Haus → 𝑗 ∈ Fre)
9897ssriv 3998 . . . 4 Haus ⊆ Fre
99 sslin 4250 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
10098, 99ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
10196, 100sstri 4004 . 2 {1o, 2o} ⊆ (On ∩ Fre)
10295, 101eqssi 4011 1 (On ∩ Fre) = {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  {csn 4630  {cpr 4632   cuni 4911  Oncon0 6385  suc csuc 6387  cfv 6562  1oc1o 8497  2oc2o 8498  Topctop 22914  Clsdccld 23039  Frect1 23330  Hauscha 23331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-1o 8504  df-2o 8505  df-topgen 17489  df-top 22915  df-topon 22932  df-cld 23042  df-t1 23337  df-haus 23338
This theorem is referenced by:  oninhaus  36432
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