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Theorem onintopssconn 33845
Description: An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.)
Assertion
Ref Expression
onintopssconn (On ∩ Top) ⊆ Conn

Proof of Theorem onintopssconn
StepHypRef Expression
1 elin 3935 . . 3 (𝑥 ∈ (On ∩ Top) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Top))
2 eloni 6188 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
3 ordtopconn 33844 . . . . 5 (Ord 𝑥 → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn))
42, 3syl 17 . . . 4 (𝑥 ∈ On → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn))
54biimpa 480 . . 3 ((𝑥 ∈ On ∧ 𝑥 ∈ Top) → 𝑥 ∈ Conn)
61, 5sylbi 220 . 2 (𝑥 ∈ (On ∩ Top) → 𝑥 ∈ Conn)
76ssriv 3957 1 (On ∩ Top) ⊆ Conn
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2115  cin 3918  wss 3919  Ord word 6177  Oncon0 6178  Topctop 21501  Conncconn 22019
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-ord 6181  df-on 6182  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-topgen 16717  df-top 21502  df-bases 21554  df-cld 21627  df-conn 22020
This theorem is referenced by: (None)
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