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Theorem ssoninhaus 32572
Description: The ordinal topologies 1𝑜 and 2𝑜 are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)
Assertion
Ref Expression
ssoninhaus {1𝑜, 2𝑜} ⊆ (On ∩ Haus)

Proof of Theorem ssoninhaus
StepHypRef Expression
1 1on 7612 . . 3 1𝑜 ∈ On
2 2on 7613 . . 3 2𝑜 ∈ On
3 prssi 4385 . . 3 ((1𝑜 ∈ On ∧ 2𝑜 ∈ On) → {1𝑜, 2𝑜} ⊆ On)
41, 2, 3mp2an 708 . 2 {1𝑜, 2𝑜} ⊆ On
5 df1o2 7617 . . . . 5 1𝑜 = {∅}
6 pw0 4375 . . . . 5 𝒫 ∅ = {∅}
75, 6eqtr4i 2676 . . . 4 1𝑜 = 𝒫 ∅
8 0ex 4823 . . . . 5 ∅ ∈ V
9 dishaus 21234 . . . . 5 (∅ ∈ V → 𝒫 ∅ ∈ Haus)
108, 9ax-mp 5 . . . 4 𝒫 ∅ ∈ Haus
117, 10eqeltri 2726 . . 3 1𝑜 ∈ Haus
12 df2o2 7619 . . . . 5 2𝑜 = {∅, {∅}}
13 pwpw0 4376 . . . . 5 𝒫 {∅} = {∅, {∅}}
1412, 13eqtr4i 2676 . . . 4 2𝑜 = 𝒫 {∅}
15 p0ex 4883 . . . . 5 {∅} ∈ V
16 dishaus 21234 . . . . 5 ({∅} ∈ V → 𝒫 {∅} ∈ Haus)
1715, 16ax-mp 5 . . . 4 𝒫 {∅} ∈ Haus
1814, 17eqeltri 2726 . . 3 2𝑜 ∈ Haus
19 prssi 4385 . . 3 ((1𝑜 ∈ Haus ∧ 2𝑜 ∈ Haus) → {1𝑜, 2𝑜} ⊆ Haus)
2011, 18, 19mp2an 708 . 2 {1𝑜, 2𝑜} ⊆ Haus
214, 20ssini 3869 1 {1𝑜, 2𝑜} ⊆ (On ∩ Haus)
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  {cpr 4212  Oncon0 5761  1𝑜c1o 7598  2𝑜c2o 7599  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-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767  df-1o 7605  df-2o 7606  df-top 20747  df-haus 21167
This theorem is referenced by:  onint1  32573  oninhaus  32574
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