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Theorem onint1 32802
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 3954 . . . . 5 (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2 eqid 2774 . . . . . . . . . . 11 𝑗 = 𝑗
32ist1 21366 . . . . . . . . . 10 (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
43simprbi 485 . . . . . . . . 9 (𝑗 ∈ Fre → ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
5 onelon 5902 . . . . . . . . . . . . . . 15 ((𝑗 ∈ On ∧ ( 𝑗 ∖ {∅}) ∈ 𝑗) → ( 𝑗 ∖ {∅}) ∈ On)
65ex 398 . . . . . . . . . . . . . 14 (𝑗 ∈ On → (( 𝑗 ∖ {∅}) ∈ 𝑗 → ( 𝑗 ∖ {∅}) ∈ On))
7 neldifsnd 4470 . . . . . . . . . . . . . . . . 17 (2𝑜𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ {∅}))
8 p0ex 4998 . . . . . . . . . . . . . . . . . . . . . 22 {∅} ∈ V
98prid2 4445 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ {∅, {∅}}
10 df2o2 7749 . . . . . . . . . . . . . . . . . . . . 21 2𝑜 = {∅, {∅}}
119, 10eleqtrri 2852 . . . . . . . . . . . . . . . . . . . 20 {∅} ∈ 2𝑜
12 elunii 4590 . . . . . . . . . . . . . . . . . . . 20 (({∅} ∈ 2𝑜 ∧ 2𝑜𝑗) → {∅} ∈ 𝑗)
1311, 12mpan 671 . . . . . . . . . . . . . . . . . . 19 (2𝑜𝑗 → {∅} ∈ 𝑗)
14 df1o2 7747 . . . . . . . . . . . . . . . . . . . . . 22 1𝑜 = {∅}
15 1on 7741 . . . . . . . . . . . . . . . . . . . . . 22 1𝑜 ∈ On
1614, 15eqeltrri 2850 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ On
1716onirri 5988 . . . . . . . . . . . . . . . . . . . 20 ¬ {∅} ∈ {∅}
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (2𝑜𝑗 → ¬ {∅} ∈ {∅})
1913, 18eldifd 3740 . . . . . . . . . . . . . . . . . 18 (2𝑜𝑗 → {∅} ∈ ( 𝑗 ∖ {∅}))
2019ne0d 4080 . . . . . . . . . . . . . . . . 17 (2𝑜𝑗 → ( 𝑗 ∖ {∅}) ≠ ∅)
217, 202thd 256 . . . . . . . . . . . . . . . 16 (2𝑜𝑗 → (¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
22 nbbn 373 . . . . . . . . . . . . . . . 16 ((¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2321, 22sylib 209 . . . . . . . . . . . . . . 15 (2𝑜𝑗 → ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
24 on0eln0 5934 . . . . . . . . . . . . . . 15 (( 𝑗 ∖ {∅}) ∈ On → (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2523, 24nsyl 137 . . . . . . . . . . . . . 14 (2𝑜𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ On)
266, 25nsyli 156 . . . . . . . . . . . . 13 (𝑗 ∈ On → (2𝑜𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗))
2726imp 394 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗)
28 0ex 4937 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
2928prid1 4444 . . . . . . . . . . . . . . . . 17 ∅ ∈ {∅, {∅}}
3029, 10eleqtrri 2852 . . . . . . . . . . . . . . . 16 ∅ ∈ 2𝑜
31 elunii 4590 . . . . . . . . . . . . . . . 16 ((∅ ∈ 2𝑜 ∧ 2𝑜𝑗) → ∅ ∈ 𝑗)
3230, 31mpan 671 . . . . . . . . . . . . . . 15 (2𝑜𝑗 → ∅ ∈ 𝑗)
3332adantl 468 . . . . . . . . . . . . . 14 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ∅ ∈ 𝑗)
34 simpr 472 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
3534sneqd 4338 . . . . . . . . . . . . . . 15 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
3635eleq1d 2838 . . . . . . . . . . . . . 14 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
3733, 36rspcdv 3468 . . . . . . . . . . . . 13 ((𝑗 ∈ On ∧ 2𝑜𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
382cldopn 21076 . . . . . . . . . . . . 13 ({∅} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗)
3937, 38syl6 35 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2𝑜𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗))
4027, 39mtod 189 . . . . . . . . . . 11 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
4140ex 398 . . . . . . . . . 10 (𝑗 ∈ On → (2𝑜𝑗 → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
4241con2d 131 . . . . . . . . 9 (𝑗 ∈ On → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ¬ 2𝑜𝑗))
434, 42syl5 34 . . . . . . . 8 (𝑗 ∈ On → (𝑗 ∈ Fre → ¬ 2𝑜𝑗))
44 2on 7743 . . . . . . . . 9 2𝑜 ∈ On
45 ontri1 5911 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (𝑗 ⊆ 2𝑜 ↔ ¬ 2𝑜𝑗))
46 onsssuc 5967 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (𝑗 ⊆ 2𝑜𝑗 ∈ suc 2𝑜))
4745, 46bitr3d 271 . . . . . . . . 9 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (¬ 2𝑜𝑗𝑗 ∈ suc 2𝑜))
4844, 47mpan2 672 . . . . . . . 8 (𝑗 ∈ On → (¬ 2𝑜𝑗𝑗 ∈ suc 2𝑜))
4943, 48sylibd 230 . . . . . . 7 (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2𝑜))
5049imp 394 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2𝑜)
51 0ntop 20950 . . . . . . . . . 10 ¬ ∅ ∈ Top
52 t1top 21375 . . . . . . . . . 10 (∅ ∈ Fre → ∅ ∈ Top)
5351, 52mto 188 . . . . . . . . 9 ¬ ∅ ∈ Fre
54 nelneq 2877 . . . . . . . . 9 ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
5553, 54mpan2 672 . . . . . . . 8 (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
56 elsni 4343 . . . . . . . 8 (𝑗 ∈ {∅} → 𝑗 = ∅)
5755, 56nsyl 137 . . . . . . 7 (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
5857adantl 468 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
5950, 58eldifd 3740 . . . . 5 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2𝑜 ∖ {∅}))
601, 59sylbi 208 . . . 4 (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2𝑜 ∖ {∅}))
6160ssriv 3762 . . 3 (On ∩ Fre) ⊆ (suc 2𝑜 ∖ {∅})
62 df-suc 5883 . . . . . 6 suc 2𝑜 = (2𝑜 ∪ {2𝑜})
6362difeq1i 3882 . . . . 5 (suc 2𝑜 ∖ {∅}) = ((2𝑜 ∪ {2𝑜}) ∖ {∅})
64 difundir 4039 . . . . 5 ((2𝑜 ∪ {2𝑜}) ∖ {∅}) = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
6563, 64eqtri 2796 . . . 4 (suc 2𝑜 ∖ {∅}) = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
66 df-pr 4329 . . . . 5 {1𝑜, 2𝑜} = ({1𝑜} ∪ {2𝑜})
67 df2o3 7748 . . . . . . . . 9 2𝑜 = {∅, 1𝑜}
68 df-pr 4329 . . . . . . . . 9 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
6967, 68eqtri 2796 . . . . . . . 8 2𝑜 = ({∅} ∪ {1𝑜})
7069difeq1i 3882 . . . . . . 7 (2𝑜 ∖ {∅}) = (({∅} ∪ {1𝑜}) ∖ {∅})
71 difundir 4039 . . . . . . 7 (({∅} ∪ {1𝑜}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅}))
72 difid 4106 . . . . . . . . 9 ({∅} ∖ {∅}) = ∅
73 1n0 7750 . . . . . . . . . . . 12 1𝑜 ≠ ∅
74 disjsn2 4395 . . . . . . . . . . . 12 (1𝑜 ≠ ∅ → ({1𝑜} ∩ {∅}) = ∅)
7573, 74ax-mp 5 . . . . . . . . . . 11 ({1𝑜} ∩ {∅}) = ∅
7675difeq2i 3883 . . . . . . . . . 10 ({1𝑜} ∖ ({1𝑜} ∩ {∅})) = ({1𝑜} ∖ ∅)
77 difin 4020 . . . . . . . . . 10 ({1𝑜} ∖ ({1𝑜} ∩ {∅})) = ({1𝑜} ∖ {∅})
78 dif0 4108 . . . . . . . . . 10 ({1𝑜} ∖ ∅) = {1𝑜}
7976, 77, 783eqtr3i 2804 . . . . . . . . 9 ({1𝑜} ∖ {∅}) = {1𝑜}
8072, 79uneq12i 3923 . . . . . . . 8 (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅})) = (∅ ∪ {1𝑜})
81 uncom 3915 . . . . . . . 8 (∅ ∪ {1𝑜}) = ({1𝑜} ∪ ∅)
82 un0 4122 . . . . . . . 8 ({1𝑜} ∪ ∅) = {1𝑜}
8380, 81, 823eqtri 2800 . . . . . . 7 (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅})) = {1𝑜}
8470, 71, 833eqtri 2800 . . . . . 6 (2𝑜 ∖ {∅}) = {1𝑜}
85 2on0 7744 . . . . . . . . 9 2𝑜 ≠ ∅
86 disjsn2 4395 . . . . . . . . 9 (2𝑜 ≠ ∅ → ({2𝑜} ∩ {∅}) = ∅)
8785, 86ax-mp 5 . . . . . . . 8 ({2𝑜} ∩ {∅}) = ∅
8887difeq2i 3883 . . . . . . 7 ({2𝑜} ∖ ({2𝑜} ∩ {∅})) = ({2𝑜} ∖ ∅)
89 difin 4020 . . . . . . 7 ({2𝑜} ∖ ({2𝑜} ∩ {∅})) = ({2𝑜} ∖ {∅})
90 dif0 4108 . . . . . . 7 ({2𝑜} ∖ ∅) = {2𝑜}
9188, 89, 903eqtr3i 2804 . . . . . 6 ({2𝑜} ∖ {∅}) = {2𝑜}
9284, 91uneq12i 3923 . . . . 5 ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅})) = ({1𝑜} ∪ {2𝑜})
9366, 92eqtr4i 2799 . . . 4 {1𝑜, 2𝑜} = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
9465, 93eqtr4i 2799 . . 3 (suc 2𝑜 ∖ {∅}) = {1𝑜, 2𝑜}
9561, 94sseqtri 3793 . 2 (On ∩ Fre) ⊆ {1𝑜, 2𝑜}
96 ssoninhaus 32801 . . 3 {1𝑜, 2𝑜} ⊆ (On ∩ Haus)
97 haust1 21397 . . . . 5 (𝑗 ∈ Haus → 𝑗 ∈ Fre)
9897ssriv 3762 . . . 4 Haus ⊆ Fre
99 sslin 3994 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
10098, 99ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
10196, 100sstri 3767 . 2 {1𝑜, 2𝑜} ⊆ (On ∩ Fre)
10295, 101eqssi 3774 1 (On ∩ Fre) = {1𝑜, 2𝑜}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 383   = wceq 1634  wcel 2148  wne 2946  wral 3064  cdif 3726  cun 3727  cin 3728  wss 3729  c0 4073  {csn 4326  {cpr 4328   cuni 4585  Oncon0 5877  suc csuc 5879  cfv 6042  1𝑜c1o 7727  2𝑜c2o 7728  Topctop 20938  Clsdccld 21061  Frect1 21352  Hauscha 21353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-ord 5880  df-on 5881  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-fv 6050  df-1o 7734  df-2o 7735  df-topgen 16332  df-top 20939  df-topon 20956  df-cld 21064  df-t1 21359  df-haus 21360
This theorem is referenced by:  oninhaus  32803
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