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Theorem oninhaus 32083
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 21061 . . . . 5 (𝑥 ∈ Haus → 𝑥 ∈ Fre)
21ssriv 3592 . . . 4 Haus ⊆ Fre
3 sslin 3822 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
42, 3ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
5 onint1 32082 . . 3 (On ∩ Fre) = {1𝑜, 2𝑜}
64, 5sseqtri 3621 . 2 (On ∩ Haus) ⊆ {1𝑜, 2𝑜}
7 ssoninhaus 32081 . 2 {1𝑜, 2𝑜} ⊆ (On ∩ Haus)
86, 7eqssi 3604 1 (On ∩ Haus) = {1𝑜, 2𝑜}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cin 3559  wss 3560  {cpr 4155  Oncon0 5685  1𝑜c1o 7499  2𝑜c2o 7500  Frect1 21016  Hauscha 21017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-1o 7506  df-2o 7507  df-topgen 16020  df-top 20616  df-topon 20618  df-cld 20728  df-t1 21023  df-haus 21024
This theorem is referenced by: (None)
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