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Theorem onintopssconn 32564
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 3829 . . 3 (𝑥 ∈ (On ∩ Top) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Top))
2 eloni 5771 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
3 ordtopconn 32563 . . . . 5 (Ord 𝑥 → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn))
42, 3syl 17 . . . 4 (𝑥 ∈ On → (𝑥 ∈ Top ↔ 𝑥 ∈ Conn))
54biimpa 500 . . 3 ((𝑥 ∈ On ∧ 𝑥 ∈ Top) → 𝑥 ∈ Conn)
61, 5sylbi 207 . 2 (𝑥 ∈ (On ∩ Top) → 𝑥 ∈ Conn)
76ssriv 3640 1 (On ∩ Top) ⊆ Conn
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2030  cin 3606  wss 3607  Ord word 5760  Oncon0 5761  Topctop 20746  Conncconn 21262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-topgen 16151  df-top 20747  df-bases 20798  df-cld 20871  df-conn 21263
This theorem is referenced by: (None)
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