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Theorem onint1 32573
 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 3829 . . . . 5 (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2 eqid 2651 . . . . . . . . . . 11 𝑗 = 𝑗
32ist1 21173 . . . . . . . . . 10 (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
43simprbi 479 . . . . . . . . 9 (𝑗 ∈ Fre → ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
5 onelon 5786 . . . . . . . . . . . . . . 15 ((𝑗 ∈ On ∧ ( 𝑗 ∖ {∅}) ∈ 𝑗) → ( 𝑗 ∖ {∅}) ∈ On)
65ex 449 . . . . . . . . . . . . . 14 (𝑗 ∈ On → (( 𝑗 ∖ {∅}) ∈ 𝑗 → ( 𝑗 ∖ {∅}) ∈ On))
7 neldifsnd 4355 . . . . . . . . . . . . . . . . 17 (2𝑜𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ {∅}))
8 p0ex 4883 . . . . . . . . . . . . . . . . . . . . . 22 {∅} ∈ V
98prid2 4330 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ {∅, {∅}}
10 df2o2 7619 . . . . . . . . . . . . . . . . . . . . 21 2𝑜 = {∅, {∅}}
119, 10eleqtrri 2729 . . . . . . . . . . . . . . . . . . . 20 {∅} ∈ 2𝑜
12 elunii 4473 . . . . . . . . . . . . . . . . . . . 20 (({∅} ∈ 2𝑜 ∧ 2𝑜𝑗) → {∅} ∈ 𝑗)
1311, 12mpan 706 . . . . . . . . . . . . . . . . . . 19 (2𝑜𝑗 → {∅} ∈ 𝑗)
14 df1o2 7617 . . . . . . . . . . . . . . . . . . . . . 22 1𝑜 = {∅}
15 1on 7612 . . . . . . . . . . . . . . . . . . . . . 22 1𝑜 ∈ On
1614, 15eqeltrri 2727 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ On
1716onirri 5872 . . . . . . . . . . . . . . . . . . . 20 ¬ {∅} ∈ {∅}
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (2𝑜𝑗 → ¬ {∅} ∈ {∅})
1913, 18eldifd 3618 . . . . . . . . . . . . . . . . . 18 (2𝑜𝑗 → {∅} ∈ ( 𝑗 ∖ {∅}))
20 ne0i 3954 . . . . . . . . . . . . . . . . . 18 ({∅} ∈ ( 𝑗 ∖ {∅}) → ( 𝑗 ∖ {∅}) ≠ ∅)
2119, 20syl 17 . . . . . . . . . . . . . . . . 17 (2𝑜𝑗 → ( 𝑗 ∖ {∅}) ≠ ∅)
227, 212thd 255 . . . . . . . . . . . . . . . 16 (2𝑜𝑗 → (¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
23 nbbn 372 . . . . . . . . . . . . . . . 16 ((¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2422, 23sylib 208 . . . . . . . . . . . . . . 15 (2𝑜𝑗 → ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
25 on0eln0 5818 . . . . . . . . . . . . . . 15 (( 𝑗 ∖ {∅}) ∈ On → (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2624, 25nsyl 135 . . . . . . . . . . . . . 14 (2𝑜𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ On)
276, 26nsyli 155 . . . . . . . . . . . . 13 (𝑗 ∈ On → (2𝑜𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗))
2827imp 444 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗)
29 0ex 4823 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
3029prid1 4329 . . . . . . . . . . . . . . . . 17 ∅ ∈ {∅, {∅}}
3130, 10eleqtrri 2729 . . . . . . . . . . . . . . . 16 ∅ ∈ 2𝑜
32 elunii 4473 . . . . . . . . . . . . . . . 16 ((∅ ∈ 2𝑜 ∧ 2𝑜𝑗) → ∅ ∈ 𝑗)
3331, 32mpan 706 . . . . . . . . . . . . . . 15 (2𝑜𝑗 → ∅ ∈ 𝑗)
3433adantl 481 . . . . . . . . . . . . . 14 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ∅ ∈ 𝑗)
35 simpr 476 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
3635sneqd 4222 . . . . . . . . . . . . . . 15 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
3736eleq1d 2715 . . . . . . . . . . . . . 14 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
3834, 37rspcdv 3343 . . . . . . . . . . . . 13 ((𝑗 ∈ On ∧ 2𝑜𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
392cldopn 20883 . . . . . . . . . . . . 13 ({∅} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗)
4038, 39syl6 35 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2𝑜𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗))
4128, 40mtod 189 . . . . . . . . . . 11 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
4241ex 449 . . . . . . . . . 10 (𝑗 ∈ On → (2𝑜𝑗 → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
4342con2d 129 . . . . . . . . 9 (𝑗 ∈ On → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ¬ 2𝑜𝑗))
444, 43syl5 34 . . . . . . . 8 (𝑗 ∈ On → (𝑗 ∈ Fre → ¬ 2𝑜𝑗))
45 2on 7613 . . . . . . . . 9 2𝑜 ∈ On
46 ontri1 5795 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (𝑗 ⊆ 2𝑜 ↔ ¬ 2𝑜𝑗))
47 onsssuc 5851 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (𝑗 ⊆ 2𝑜𝑗 ∈ suc 2𝑜))
4846, 47bitr3d 270 . . . . . . . . 9 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (¬ 2𝑜𝑗𝑗 ∈ suc 2𝑜))
4945, 48mpan2 707 . . . . . . . 8 (𝑗 ∈ On → (¬ 2𝑜𝑗𝑗 ∈ suc 2𝑜))
5044, 49sylibd 229 . . . . . . 7 (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2𝑜))
5150imp 444 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2𝑜)
52 0ntop 20758 . . . . . . . . . 10 ¬ ∅ ∈ Top
53 t1top 21182 . . . . . . . . . 10 (∅ ∈ Fre → ∅ ∈ Top)
5452, 53mto 188 . . . . . . . . 9 ¬ ∅ ∈ Fre
55 nelneq 2754 . . . . . . . . 9 ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
5654, 55mpan2 707 . . . . . . . 8 (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
57 elsni 4227 . . . . . . . 8 (𝑗 ∈ {∅} → 𝑗 = ∅)
5856, 57nsyl 135 . . . . . . 7 (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
5958adantl 481 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
6051, 59eldifd 3618 . . . . 5 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2𝑜 ∖ {∅}))
611, 60sylbi 207 . . . 4 (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2𝑜 ∖ {∅}))
6261ssriv 3640 . . 3 (On ∩ Fre) ⊆ (suc 2𝑜 ∖ {∅})
63 df-suc 5767 . . . . . 6 suc 2𝑜 = (2𝑜 ∪ {2𝑜})
6463difeq1i 3757 . . . . 5 (suc 2𝑜 ∖ {∅}) = ((2𝑜 ∪ {2𝑜}) ∖ {∅})
65 difundir 3913 . . . . 5 ((2𝑜 ∪ {2𝑜}) ∖ {∅}) = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
6664, 65eqtri 2673 . . . 4 (suc 2𝑜 ∖ {∅}) = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
67 df-pr 4213 . . . . 5 {1𝑜, 2𝑜} = ({1𝑜} ∪ {2𝑜})
68 df2o3 7618 . . . . . . . . 9 2𝑜 = {∅, 1𝑜}
69 df-pr 4213 . . . . . . . . 9 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
7068, 69eqtri 2673 . . . . . . . 8 2𝑜 = ({∅} ∪ {1𝑜})
7170difeq1i 3757 . . . . . . 7 (2𝑜 ∖ {∅}) = (({∅} ∪ {1𝑜}) ∖ {∅})
72 difundir 3913 . . . . . . 7 (({∅} ∪ {1𝑜}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅}))
73 difid 3981 . . . . . . . . 9 ({∅} ∖ {∅}) = ∅
74 1n0 7620 . . . . . . . . . . . 12 1𝑜 ≠ ∅
75 disjsn2 4279 . . . . . . . . . . . 12 (1𝑜 ≠ ∅ → ({1𝑜} ∩ {∅}) = ∅)
7674, 75ax-mp 5 . . . . . . . . . . 11 ({1𝑜} ∩ {∅}) = ∅
7776difeq2i 3758 . . . . . . . . . 10 ({1𝑜} ∖ ({1𝑜} ∩ {∅})) = ({1𝑜} ∖ ∅)
78 difin 3894 . . . . . . . . . 10 ({1𝑜} ∖ ({1𝑜} ∩ {∅})) = ({1𝑜} ∖ {∅})
79 dif0 3983 . . . . . . . . . 10 ({1𝑜} ∖ ∅) = {1𝑜}
8077, 78, 793eqtr3i 2681 . . . . . . . . 9 ({1𝑜} ∖ {∅}) = {1𝑜}
8173, 80uneq12i 3798 . . . . . . . 8 (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅})) = (∅ ∪ {1𝑜})
82 uncom 3790 . . . . . . . 8 (∅ ∪ {1𝑜}) = ({1𝑜} ∪ ∅)
83 un0 4000 . . . . . . . 8 ({1𝑜} ∪ ∅) = {1𝑜}
8481, 82, 833eqtri 2677 . . . . . . 7 (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅})) = {1𝑜}
8571, 72, 843eqtri 2677 . . . . . 6 (2𝑜 ∖ {∅}) = {1𝑜}
86 2on0 7614 . . . . . . . . 9 2𝑜 ≠ ∅
87 disjsn2 4279 . . . . . . . . 9 (2𝑜 ≠ ∅ → ({2𝑜} ∩ {∅}) = ∅)
8886, 87ax-mp 5 . . . . . . . 8 ({2𝑜} ∩ {∅}) = ∅
8988difeq2i 3758 . . . . . . 7 ({2𝑜} ∖ ({2𝑜} ∩ {∅})) = ({2𝑜} ∖ ∅)
90 difin 3894 . . . . . . 7 ({2𝑜} ∖ ({2𝑜} ∩ {∅})) = ({2𝑜} ∖ {∅})
91 dif0 3983 . . . . . . 7 ({2𝑜} ∖ ∅) = {2𝑜}
9289, 90, 913eqtr3i 2681 . . . . . 6 ({2𝑜} ∖ {∅}) = {2𝑜}
9385, 92uneq12i 3798 . . . . 5 ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅})) = ({1𝑜} ∪ {2𝑜})
9467, 93eqtr4i 2676 . . . 4 {1𝑜, 2𝑜} = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
9566, 94eqtr4i 2676 . . 3 (suc 2𝑜 ∖ {∅}) = {1𝑜, 2𝑜}
9662, 95sseqtri 3670 . 2 (On ∩ Fre) ⊆ {1𝑜, 2𝑜}
97 ssoninhaus 32572 . . 3 {1𝑜, 2𝑜} ⊆ (On ∩ Haus)
98 haust1 21204 . . . . 5 (𝑗 ∈ Haus → 𝑗 ∈ Fre)
9998ssriv 3640 . . . 4 Haus ⊆ Fre
100 sslin 3872 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
10199, 100ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
10297, 101sstri 3645 . 2 {1𝑜, 2𝑜} ⊆ (On ∩ Fre)
10396, 102eqssi 3652 1 (On ∩ Fre) = {1𝑜, 2𝑜}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  {csn 4210  {cpr 4212  ∪ cuni 4468  Oncon0 5761  suc csuc 5763  ‘cfv 5926  1𝑜c1o 7598  2𝑜c2o 7599  Topctop 20746  Clsdccld 20868  Frect1 21159  Hauscha 21160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-1o 7605  df-2o 7606  df-topgen 16151  df-top 20747  df-topon 20764  df-cld 20871  df-t1 21166  df-haus 21167 This theorem is referenced by:  oninhaus  32574
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