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Theorem oninhaus 33355
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 21679 . . . . 5 (𝑥 ∈ Haus → 𝑥 ∈ Fre)
21ssriv 3855 . . . 4 Haus ⊆ Fre
3 sslin 4092 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
42, 3ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
5 onint1 33354 . . 3 (On ∩ Fre) = {1o, 2o}
64, 5sseqtri 3886 . 2 (On ∩ Haus) ⊆ {1o, 2o}
7 ssoninhaus 33353 . 2 {1o, 2o} ⊆ (On ∩ Haus)
86, 7eqssi 3867 1 (On ∩ Haus) = {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1508  cin 3821  wss 3822  {cpr 4437  Oncon0 6026  1oc1o 7896  2oc2o 7897  Frect1 21634  Hauscha 21635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-ord 6029  df-on 6030  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-fv 6193  df-1o 7903  df-2o 7904  df-topgen 16571  df-top 21221  df-topon 21238  df-cld 21346  df-t1 21641  df-haus 21642
This theorem is referenced by: (None)
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