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Theorem onint1 31424
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 3757 . . . . 5 (𝑗 ∈ (On ∩ Fre) ↔ (𝑗 ∈ On ∧ 𝑗 ∈ Fre))
2 eqid 2609 . . . . . . . . . . 11 𝑗 = 𝑗
32ist1 20877 . . . . . . . . . 10 (𝑗 ∈ Fre ↔ (𝑗 ∈ Top ∧ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
43simprbi 478 . . . . . . . . 9 (𝑗 ∈ Fre → ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
5 onelon 5651 . . . . . . . . . . . . . . 15 ((𝑗 ∈ On ∧ ( 𝑗 ∖ {∅}) ∈ 𝑗) → ( 𝑗 ∖ {∅}) ∈ On)
65ex 448 . . . . . . . . . . . . . 14 (𝑗 ∈ On → (( 𝑗 ∖ {∅}) ∈ 𝑗 → ( 𝑗 ∖ {∅}) ∈ On))
7 neldifsnd 4262 . . . . . . . . . . . . . . . . 17 (2𝑜𝑗 → ¬ ∅ ∈ ( 𝑗 ∖ {∅}))
8 p0ex 4774 . . . . . . . . . . . . . . . . . . . . . 22 {∅} ∈ V
98prid2 4241 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ {∅, {∅}}
10 df2o2 7438 . . . . . . . . . . . . . . . . . . . . 21 2𝑜 = {∅, {∅}}
119, 10eleqtrri 2686 . . . . . . . . . . . . . . . . . . . 20 {∅} ∈ 2𝑜
12 elunii 4371 . . . . . . . . . . . . . . . . . . . 20 (({∅} ∈ 2𝑜 ∧ 2𝑜𝑗) → {∅} ∈ 𝑗)
1311, 12mpan 701 . . . . . . . . . . . . . . . . . . 19 (2𝑜𝑗 → {∅} ∈ 𝑗)
14 df1o2 7436 . . . . . . . . . . . . . . . . . . . . . 22 1𝑜 = {∅}
15 1on 7431 . . . . . . . . . . . . . . . . . . . . . 22 1𝑜 ∈ On
1614, 15eqeltrri 2684 . . . . . . . . . . . . . . . . . . . . 21 {∅} ∈ On
1716onirri 5737 . . . . . . . . . . . . . . . . . . . 20 ¬ {∅} ∈ {∅}
1817a1i 11 . . . . . . . . . . . . . . . . . . 19 (2𝑜𝑗 → ¬ {∅} ∈ {∅})
1913, 18eldifd 3550 . . . . . . . . . . . . . . . . . 18 (2𝑜𝑗 → {∅} ∈ ( 𝑗 ∖ {∅}))
20 ne0i 3879 . . . . . . . . . . . . . . . . . 18 ({∅} ∈ ( 𝑗 ∖ {∅}) → ( 𝑗 ∖ {∅}) ≠ ∅)
2119, 20syl 17 . . . . . . . . . . . . . . . . 17 (2𝑜𝑗 → ( 𝑗 ∖ {∅}) ≠ ∅)
227, 212thd 253 . . . . . . . . . . . . . . . 16 (2𝑜𝑗 → (¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
23 nbbn 371 . . . . . . . . . . . . . . . 16 ((¬ ∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅) ↔ ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2422, 23sylib 206 . . . . . . . . . . . . . . 15 (2𝑜𝑗 → ¬ (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
25 on0eln0 5683 . . . . . . . . . . . . . . 15 (( 𝑗 ∖ {∅}) ∈ On → (∅ ∈ ( 𝑗 ∖ {∅}) ↔ ( 𝑗 ∖ {∅}) ≠ ∅))
2624, 25nsyl 133 . . . . . . . . . . . . . 14 (2𝑜𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ On)
276, 26nsyli 153 . . . . . . . . . . . . 13 (𝑗 ∈ On → (2𝑜𝑗 → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗))
2827imp 443 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ¬ ( 𝑗 ∖ {∅}) ∈ 𝑗)
29 0ex 4713 . . . . . . . . . . . . . . . . . 18 ∅ ∈ V
3029prid1 4240 . . . . . . . . . . . . . . . . 17 ∅ ∈ {∅, {∅}}
3130, 10eleqtrri 2686 . . . . . . . . . . . . . . . 16 ∅ ∈ 2𝑜
32 elunii 4371 . . . . . . . . . . . . . . . 16 ((∅ ∈ 2𝑜 ∧ 2𝑜𝑗) → ∅ ∈ 𝑗)
3331, 32mpan 701 . . . . . . . . . . . . . . 15 (2𝑜𝑗 → ∅ ∈ 𝑗)
3433adantl 480 . . . . . . . . . . . . . 14 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ∅ ∈ 𝑗)
35 simpr 475 . . . . . . . . . . . . . . . 16 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → 𝑎 = ∅)
3635sneqd 4136 . . . . . . . . . . . . . . 15 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → {𝑎} = {∅})
3736eleq1d 2671 . . . . . . . . . . . . . 14 (((𝑗 ∈ On ∧ 2𝑜𝑗) ∧ 𝑎 = ∅) → ({𝑎} ∈ (Clsd‘𝑗) ↔ {∅} ∈ (Clsd‘𝑗)))
3834, 37rspcdv 3284 . . . . . . . . . . . . 13 ((𝑗 ∈ On ∧ 2𝑜𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → {∅} ∈ (Clsd‘𝑗)))
392cldopn 20587 . . . . . . . . . . . . 13 ({∅} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗)
4038, 39syl6 34 . . . . . . . . . . . 12 ((𝑗 ∈ On ∧ 2𝑜𝑗) → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ( 𝑗 ∖ {∅}) ∈ 𝑗))
4128, 40mtod 187 . . . . . . . . . . 11 ((𝑗 ∈ On ∧ 2𝑜𝑗) → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗))
4241ex 448 . . . . . . . . . 10 (𝑗 ∈ On → (2𝑜𝑗 → ¬ ∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗)))
4342con2d 127 . . . . . . . . 9 (𝑗 ∈ On → (∀𝑎 𝑗{𝑎} ∈ (Clsd‘𝑗) → ¬ 2𝑜𝑗))
444, 43syl5 33 . . . . . . . 8 (𝑗 ∈ On → (𝑗 ∈ Fre → ¬ 2𝑜𝑗))
45 2on 7432 . . . . . . . . 9 2𝑜 ∈ On
46 ontri1 5660 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (𝑗 ⊆ 2𝑜 ↔ ¬ 2𝑜𝑗))
47 onsssuc 5716 . . . . . . . . . 10 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (𝑗 ⊆ 2𝑜𝑗 ∈ suc 2𝑜))
4846, 47bitr3d 268 . . . . . . . . 9 ((𝑗 ∈ On ∧ 2𝑜 ∈ On) → (¬ 2𝑜𝑗𝑗 ∈ suc 2𝑜))
4945, 48mpan2 702 . . . . . . . 8 (𝑗 ∈ On → (¬ 2𝑜𝑗𝑗 ∈ suc 2𝑜))
5044, 49sylibd 227 . . . . . . 7 (𝑗 ∈ On → (𝑗 ∈ Fre → 𝑗 ∈ suc 2𝑜))
5150imp 443 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ suc 2𝑜)
52 0ntop 20477 . . . . . . . . . 10 ¬ ∅ ∈ Top
53 t1top 20886 . . . . . . . . . 10 (∅ ∈ Fre → ∅ ∈ Top)
5452, 53mto 186 . . . . . . . . 9 ¬ ∅ ∈ Fre
55 nelneq 2711 . . . . . . . . 9 ((𝑗 ∈ Fre ∧ ¬ ∅ ∈ Fre) → ¬ 𝑗 = ∅)
5654, 55mpan2 702 . . . . . . . 8 (𝑗 ∈ Fre → ¬ 𝑗 = ∅)
57 elsni 4141 . . . . . . . 8 (𝑗 ∈ {∅} → 𝑗 = ∅)
5856, 57nsyl 133 . . . . . . 7 (𝑗 ∈ Fre → ¬ 𝑗 ∈ {∅})
5958adantl 480 . . . . . 6 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → ¬ 𝑗 ∈ {∅})
6051, 59eldifd 3550 . . . . 5 ((𝑗 ∈ On ∧ 𝑗 ∈ Fre) → 𝑗 ∈ (suc 2𝑜 ∖ {∅}))
611, 60sylbi 205 . . . 4 (𝑗 ∈ (On ∩ Fre) → 𝑗 ∈ (suc 2𝑜 ∖ {∅}))
6261ssriv 3571 . . 3 (On ∩ Fre) ⊆ (suc 2𝑜 ∖ {∅})
63 df-suc 5632 . . . . . 6 suc 2𝑜 = (2𝑜 ∪ {2𝑜})
6463difeq1i 3685 . . . . 5 (suc 2𝑜 ∖ {∅}) = ((2𝑜 ∪ {2𝑜}) ∖ {∅})
65 difundir 3838 . . . . 5 ((2𝑜 ∪ {2𝑜}) ∖ {∅}) = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
6664, 65eqtri 2631 . . . 4 (suc 2𝑜 ∖ {∅}) = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
67 df-pr 4127 . . . . 5 {1𝑜, 2𝑜} = ({1𝑜} ∪ {2𝑜})
68 df2o3 7437 . . . . . . . . 9 2𝑜 = {∅, 1𝑜}
69 df-pr 4127 . . . . . . . . 9 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
7068, 69eqtri 2631 . . . . . . . 8 2𝑜 = ({∅} ∪ {1𝑜})
7170difeq1i 3685 . . . . . . 7 (2𝑜 ∖ {∅}) = (({∅} ∪ {1𝑜}) ∖ {∅})
72 difundir 3838 . . . . . . 7 (({∅} ∪ {1𝑜}) ∖ {∅}) = (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅}))
73 difid 3901 . . . . . . . . 9 ({∅} ∖ {∅}) = ∅
74 1n0 7439 . . . . . . . . . . . 12 1𝑜 ≠ ∅
75 disjsn2 4192 . . . . . . . . . . . 12 (1𝑜 ≠ ∅ → ({1𝑜} ∩ {∅}) = ∅)
7674, 75ax-mp 5 . . . . . . . . . . 11 ({1𝑜} ∩ {∅}) = ∅
7776difeq2i 3686 . . . . . . . . . 10 ({1𝑜} ∖ ({1𝑜} ∩ {∅})) = ({1𝑜} ∖ ∅)
78 difin 3822 . . . . . . . . . 10 ({1𝑜} ∖ ({1𝑜} ∩ {∅})) = ({1𝑜} ∖ {∅})
79 dif0 3903 . . . . . . . . . 10 ({1𝑜} ∖ ∅) = {1𝑜}
8077, 78, 793eqtr3i 2639 . . . . . . . . 9 ({1𝑜} ∖ {∅}) = {1𝑜}
8173, 80uneq12i 3726 . . . . . . . 8 (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅})) = (∅ ∪ {1𝑜})
82 uncom 3718 . . . . . . . 8 (∅ ∪ {1𝑜}) = ({1𝑜} ∪ ∅)
83 un0 3918 . . . . . . . 8 ({1𝑜} ∪ ∅) = {1𝑜}
8481, 82, 833eqtri 2635 . . . . . . 7 (({∅} ∖ {∅}) ∪ ({1𝑜} ∖ {∅})) = {1𝑜}
8571, 72, 843eqtri 2635 . . . . . 6 (2𝑜 ∖ {∅}) = {1𝑜}
86 2on0 7433 . . . . . . . . 9 2𝑜 ≠ ∅
87 disjsn2 4192 . . . . . . . . 9 (2𝑜 ≠ ∅ → ({2𝑜} ∩ {∅}) = ∅)
8886, 87ax-mp 5 . . . . . . . 8 ({2𝑜} ∩ {∅}) = ∅
8988difeq2i 3686 . . . . . . 7 ({2𝑜} ∖ ({2𝑜} ∩ {∅})) = ({2𝑜} ∖ ∅)
90 difin 3822 . . . . . . 7 ({2𝑜} ∖ ({2𝑜} ∩ {∅})) = ({2𝑜} ∖ {∅})
91 dif0 3903 . . . . . . 7 ({2𝑜} ∖ ∅) = {2𝑜}
9289, 90, 913eqtr3i 2639 . . . . . 6 ({2𝑜} ∖ {∅}) = {2𝑜}
9385, 92uneq12i 3726 . . . . 5 ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅})) = ({1𝑜} ∪ {2𝑜})
9467, 93eqtr4i 2634 . . . 4 {1𝑜, 2𝑜} = ((2𝑜 ∖ {∅}) ∪ ({2𝑜} ∖ {∅}))
9566, 94eqtr4i 2634 . . 3 (suc 2𝑜 ∖ {∅}) = {1𝑜, 2𝑜}
9662, 95sseqtri 3599 . 2 (On ∩ Fre) ⊆ {1𝑜, 2𝑜}
97 ssoninhaus 31423 . . 3 {1𝑜, 2𝑜} ⊆ (On ∩ Haus)
98 haust1 20908 . . . . 5 (𝑗 ∈ Haus → 𝑗 ∈ Fre)
9998ssriv 3571 . . . 4 Haus ⊆ Fre
100 sslin 3800 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
10199, 100ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
10297, 101sstri 3576 . 2 {1𝑜, 2𝑜} ⊆ (On ∩ Fre)
10396, 102eqssi 3583 1 (On ∩ Fre) = {1𝑜, 2𝑜}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779  wral 2895  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124  {cpr 4126   cuni 4366  Oncon0 5626  suc csuc 5628  cfv 5790  1𝑜c1o 7417  2𝑜c2o 7418  Topctop 20459  Clsdccld 20572  Frect1 20863  Hauscha 20864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-ord 5629  df-on 5630  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798  df-1o 7424  df-2o 7425  df-topgen 15873  df-top 20463  df-topon 20465  df-cld 20575  df-t1 20870  df-haus 20871
This theorem is referenced by:  oninhaus  31425
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