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Theorem onintopssconn 33685
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 4166 . . 3 (𝑥 ∈ (On ∩ Top) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Top))
2 eloni 6194 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
3 ordtopconn 33684 . . . . 5 (Ord 𝑥 → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn))
42, 3syl 17 . . . 4 (𝑥 ∈ On → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn))
54biimpa 477 . . 3 ((𝑥 ∈ On ∧ 𝑥 ∈ Top) → 𝑥 ∈ Conn)
61, 5sylbi 218 . 2 (𝑥 ∈ (On ∩ Top) → 𝑥 ∈ Conn)
76ssriv 3968 1 (On ∩ Top) ⊆ Conn
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  cin 3932  wss 3933  Ord word 6183  Oncon0 6184  Topctop 21429  Conncconn 21947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-topgen 16705  df-top 21430  df-bases 21482  df-cld 21555  df-conn 21948
This theorem is referenced by: (None)
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