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Theorem onint1 36174
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 3962 . . . . 5 (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2 eqid 2726 . . . . . . . . . . 11 𝑗 = 𝑗
32ist1 23313 . . . . . . . . . 10 (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
43simprbi 495 . . . . . . . . 9 (𝑗 ∈ Fre → ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
5 onelon 6393 . . . . . . . . . . . . . . 15 ((𝑗 ∈ On ∧ ( 𝑗 ∖ {∅}) ∈ 𝑗) → ( 𝑗 ∖ {∅}) ∈ On)
65ex 411 . . . . . . . . . . . . . 14 (𝑗 ∈ On → (( 𝑗 ∖ {∅}) ∈ 𝑗 → ( 𝑗 ∖ {∅}) ∈ On))
7 neldifsnd 4792 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ {∅}))
8 p0ex 5380 . . . . . . . . . . . . . . . . . . . . . 22 {∅} ∈ V
98prid2 4762 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ {∅, {∅}}
10 df2o2 8497 . . . . . . . . . . . . . . . . . . . . 21 2o = {∅, {∅}}
119, 10eleqtrri 2825 . . . . . . . . . . . . . . . . . . . 20 {∅} ∈ 2o
12 elunii 4910 . . . . . . . . . . . . . . . . . . . 20 (({∅} ∈ 2o ∧ 2o𝑗) → {∅} ∈ 𝑗)
1311, 12mpan 688 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → {∅} ∈ 𝑗)
14 df1o2 8495 . . . . . . . . . . . . . . . . . . . . . 22 1o = {∅}
15 1on 8500 . . . . . . . . . . . . . . . . . . . . . 22 1o ∈ On
1614, 15eqeltrri 2823 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ On
1716onirri 6481 . . . . . . . . . . . . . . . . . . . 20 ¬ {∅} ∈ {∅}
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (2o𝑗 → ¬ {∅} ∈ {∅})
1913, 18eldifd 3957 . . . . . . . . . . . . . . . . . 18 (2o𝑗 → {∅} ∈ ( 𝑗 ∖ {∅}))
2019ne0d 4335 . . . . . . . . . . . . . . . . 17 (2o𝑗 → ( 𝑗 ∖ {∅}) ≠ ∅)
217, 202thd 264 . . . . . . . . . . . . . . . 16 (2o𝑗 → (¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
22 nbbn 382 . . . . . . . . . . . . . . . 16 ((¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2321, 22sylib 217 . . . . . . . . . . . . . . 15 (2o𝑗 → ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
24 on0eln0 6424 . . . . . . . . . . . . . . 15 (( 𝑗 ∖ {∅}) ∈ On → (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2523, 24nsyl 140 . . . . . . . . . . . . . 14 (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ On)
266, 25nsyli 157 . . . . . . . . . . . . 13 (𝑗 ∈ On → (2o𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗))
2726imp 405 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗)
28 0ex 5304 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2928prid1 4761 . . . . . . . . . . . . . . . . 17 ∅ ∈ {∅, {∅}}
3029, 10eleqtrri 2825 . . . . . . . . . . . . . . . 16 ∅ ∈ 2o
31 elunii 4910 . . . . . . . . . . . . . . . 16 ((∅ ∈ 2o ∧ 2o𝑗) → ∅ ∈ 𝑗)
3230, 31mpan 688 . . . . . . . . . . . . . . 15 (2o𝑗 → ∅ ∈ 𝑗)
3332adantl 480 . . . . . . . . . . . . . 14 ((𝑗 ∈ On ∧ 2o𝑗) → ∅ ∈ 𝑗)
34 simpr 483 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
3534sneqd 4635 . . . . . . . . . . . . . . 15 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
3635eleq1d 2811 . . . . . . . . . . . . . 14 (((𝑗 ∈ On ∧ 2o𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
3733, 36rspcdv 3599 . . . . . . . . . . . . 13 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
382cldopn 23023 . . . . . . . . . . . . 13 ({∅} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗)
3937, 38syl6 35 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2o𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗))
4027, 39mtod 197 . . . . . . . . . . 11 ((𝑗 ∈ On ∧ 2o𝑗) → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
4140ex 411 . . . . . . . . . 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 8502 . . . . . . . . 9 2o ∈ On
45 ontri1 6402 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o ↔ ¬ 2o𝑗))
46 onsssuc 6458 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2o ∈ On) → (𝑗 ⊆ 2o𝑗 ∈ suc 2o))
4745, 46bitr3d 280 . . . . . . . . 9 ((𝑗 ∈ On ∧ 2o ∈ On) → (¬ 2o𝑗𝑗 ∈ suc 2o))
4844, 47mpan2 689 . . . . . . . 8 (𝑗 ∈ On → (¬ 2o𝑗𝑗 ∈ suc 2o))
4943, 48sylibd 238 . . . . . . 7 (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2o))
5049imp 405 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2o)
51 0ntop 22895 . . . . . . . . . 10 ¬ ∅ ∈ Top
52 t1top 23322 . . . . . . . . . 10 (∅ ∈ Fre → ∅ ∈ Top)
5351, 52mto 196 . . . . . . . . 9 ¬ ∅ ∈ Fre
54 nelneq 2850 . . . . . . . . 9 ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
5553, 54mpan2 689 . . . . . . . 8 (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
56 elsni 4640 . . . . . . . 8 (𝑗 ∈ {∅} → 𝑗 = ∅)
5755, 56nsyl 140 . . . . . . 7 (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
5857adantl 480 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
5950, 58eldifd 3957 . . . . 5 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
601, 59sylbi 216 . . . 4 (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2o ∖ {∅}))
6160ssriv 3982 . . 3 (On ∩ Fre) ⊆ (suc 2o ∖ {∅})
62 df-suc 6374 . . . . . 6 suc 2o = (2o ∪ {2o})
6362difeq1i 4114 . . . . 5 (suc 2o ∖ {∅}) = ((2o ∪ {2o}) ∖ {∅})
64 difundir 4279 . . . . 5 ((2o ∪ {2o}) ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
6563, 64eqtri 2754 . . . 4 (suc 2o ∖ {∅}) = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
66 df-pr 4626 . . . . 5 {1o, 2o} = ({1o} ∪ {2o})
67 df2o3 8496 . . . . . . . . 9 2o = {∅, 1o}
68 df-pr 4626 . . . . . . . . 9 {∅, 1o} = ({∅} ∪ {1o})
6967, 68eqtri 2754 . . . . . . . 8 2o = ({∅} ∪ {1o})
7069difeq1i 4114 . . . . . . 7 (2o ∖ {∅}) = (({∅} ∪ {1o}) ∖ {∅})
71 difundir 4279 . . . . . . 7 (({∅} ∪ {1o}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅}))
72 difid 4368 . . . . . . . . 9 ({∅} ∖ {∅}) = ∅
73 1n0 8510 . . . . . . . . . . . 12 1o ≠ ∅
74 disjsn2 4711 . . . . . . . . . . . 12 (1o ≠ ∅ → ({1o} ∩ {∅}) = ∅)
7573, 74ax-mp 5 . . . . . . . . . . 11 ({1o} ∩ {∅}) = ∅
7675difeq2i 4115 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ ∅)
77 difin 4260 . . . . . . . . . 10 ({1o} ∖ ({1o} ∩ {∅})) = ({1o} ∖ {∅})
78 dif0 4370 . . . . . . . . . 10 ({1o} ∖ ∅) = {1o}
7976, 77, 783eqtr3i 2762 . . . . . . . . 9 ({1o} ∖ {∅}) = {1o}
8072, 79uneq12i 4158 . . . . . . . 8 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = (∅ ∪ {1o})
81 uncom 4150 . . . . . . . 8 (∅ ∪ {1o}) = ({1o} ∪ ∅)
82 un0 4388 . . . . . . . 8 ({1o} ∪ ∅) = {1o}
8380, 81, 823eqtri 2758 . . . . . . 7 (({∅} ∖ {∅}) ∪ ({1o} ∖ {∅})) = {1o}
8470, 71, 833eqtri 2758 . . . . . 6 (2o ∖ {∅}) = {1o}
85 2on0 8504 . . . . . . . . 9 2o ≠ ∅
86 disjsn2 4711 . . . . . . . . 9 (2o ≠ ∅ → ({2o} ∩ {∅}) = ∅)
8785, 86ax-mp 5 . . . . . . . 8 ({2o} ∩ {∅}) = ∅
8887difeq2i 4115 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ ∅)
89 difin 4260 . . . . . . 7 ({2o} ∖ ({2o} ∩ {∅})) = ({2o} ∖ {∅})
90 dif0 4370 . . . . . . 7 ({2o} ∖ ∅) = {2o}
9188, 89, 903eqtr3i 2762 . . . . . 6 ({2o} ∖ {∅}) = {2o}
9284, 91uneq12i 4158 . . . . 5 ((2o ∖ {∅}) ∪ ({2o} ∖ {∅})) = ({1o} ∪ {2o})
9366, 92eqtr4i 2757 . . . 4 {1o, 2o} = ((2o ∖ {∅}) ∪ ({2o} ∖ {∅}))
9465, 93eqtr4i 2757 . . 3 (suc 2o ∖ {∅}) = {1o, 2o}
9561, 94sseqtri 4015 . 2 (On ∩ Fre) ⊆ {1o, 2o}
96 ssoninhaus 36173 . . 3 {1o, 2o} ⊆ (On ∩ Haus)
97 haust1 23344 . . . . 5 (𝑗 ∈ Haus → 𝑗 ∈ Fre)
9897ssriv 3982 . . . 4 Haus ⊆ Fre
99 sslin 4233 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
10098, 99ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
10196, 100sstri 3988 . 2 {1o, 2o} ⊆ (On ∩ Fre)
10295, 101eqssi 3995 1 (On ∩ Fre) = {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  cdif 3943  cun 3944  cin 3945  wss 3946  c0 4322  {csn 4623  {cpr 4625   cuni 4905  Oncon0 6368  suc csuc 6370  cfv 6546  1oc1o 8481  2oc2o 8482  Topctop 22883  Clsdccld 23008  Frect1 23299  Hauscha 23300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-fv 6554  df-1o 8488  df-2o 8489  df-topgen 17453  df-top 22884  df-topon 22901  df-cld 23011  df-t1 23306  df-haus 23307
This theorem is referenced by:  oninhaus  36175
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