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Theorem oninhaus 36452
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 23361 . . . . 5 (𝑥 ∈ Haus → 𝑥 ∈ Fre)
21ssriv 3986 . . . 4 Haus ⊆ Fre
3 sslin 4242 . . . 4 (Haus ⊆ Fre → (On ∩ Haus) ⊆ (On ∩ Fre))
42, 3ax-mp 5 . . 3 (On ∩ Haus) ⊆ (On ∩ Fre)
5 onint1 36451 . . 3 (On ∩ Fre) = {1o, 2o}
64, 5sseqtri 4031 . 2 (On ∩ Haus) ⊆ {1o, 2o}
7 ssoninhaus 36450 . 2 {1o, 2o} ⊆ (On ∩ Haus)
86, 7eqssi 3999 1 (On ∩ Haus) = {1o, 2o}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cin 3949  wss 3950  {cpr 4627  Oncon0 6383  1oc1o 8500  2oc2o 8501  Frect1 23316  Hauscha 23317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-1o 8507  df-2o 8508  df-topgen 17489  df-top 22901  df-topon 22918  df-cld 23028  df-t1 23323  df-haus 23324
This theorem is referenced by: (None)
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