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Theorem oninhaus 33872
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 21955 . . . . 5 (𝑥 ∈ Haus → 𝑥 ∈ Fre)
21ssriv 3946 . . . 4 Haus ⊆ Fre
3 sslin 4185 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
42, 3ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
5 onint1 33871 . . 3 (On ∩ Fre) = {1o, 2o}
64, 5sseqtri 3978 . 2 (On ∩ Haus) ⊆ {1o, 2o}
7 ssoninhaus 33870 . 2 {1o, 2o} ⊆ (On ∩ Haus)
86, 7eqssi 3958 1 (On ∩ Haus) = {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cin 3907  wss 3908  {cpr 4541  Oncon0 6169  1oc1o 8082  2oc2o 8083  Frect1 21910  Hauscha 21911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342  df-1o 8089  df-2o 8090  df-topgen 16708  df-top 21497  df-topon 21514  df-cld 21622  df-t1 21917  df-haus 21918
This theorem is referenced by: (None)
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