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Theorem oninhaus 31619
Description: The ordinal Hausdorff spaces are 1𝑜 and 2𝑜. (Contributed by Chen-Pang He, 10-Nov-2015.)
Assertion
Ref Expression
oninhaus (On ∩ Haus) = {1𝑜, 2𝑜}

Proof of Theorem oninhaus
StepHypRef Expression
1 haust1 20966 . . . . 5 (𝑥 ∈ Haus → 𝑥 ∈ Fre)
21ssriv 3572 . . . 4 Haus ⊆ Fre
3 sslin 3801 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
42, 3ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
5 onint1 31618 . . 3 (On ∩ Fre) = {1𝑜, 2𝑜}
64, 5sseqtri 3600 . 2 (On ∩ Haus) ⊆ {1𝑜, 2𝑜}
7 ssoninhaus 31617 . 2 {1𝑜, 2𝑜} ⊆ (On ∩ Haus)
86, 7eqssi 3584 1 (On ∩ Haus) = {1𝑜, 2𝑜}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  cin 3539  wss 3540  {cpr 4127  Oncon0 5640  1𝑜c1o 7440  2𝑜c2o 7441  Frect1 20921  Hauscha 20922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-1o 7447  df-2o 7448  df-topgen 15927  df-top 20521  df-topon 20523  df-cld 20633  df-t1 20928  df-haus 20929
This theorem is referenced by: (None)
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