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Theorem iscon 21026
 Description: The predicate 𝐽 is a connected topology . (Contributed by FL, 17-Nov-2008.)
Hypothesis
Ref Expression
iscon.1 𝑋 = 𝐽
Assertion
Ref Expression
iscon (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))

Proof of Theorem iscon
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑗 = 𝐽𝑗 = 𝐽)
2 fveq2 6103 . . . 4 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
31, 2ineq12d 3777 . . 3 (𝑗 = 𝐽 → (𝑗 ∩ (Clsd‘𝑗)) = (𝐽 ∩ (Clsd‘𝐽)))
4 unieq 4380 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
5 iscon.1 . . . . 5 𝑋 = 𝐽
64, 5syl6eqr 2662 . . . 4 (𝑗 = 𝐽 𝑗 = 𝑋)
76preq2d 4219 . . 3 (𝑗 = 𝐽 → {∅, 𝑗} = {∅, 𝑋})
83, 7eqeq12d 2625 . 2 (𝑗 = 𝐽 → ((𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗} ↔ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9 df-con 21025 . 2 Con = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, 𝑗}}
108, 9elrab2 3333 1 (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ∅c0 3874  {cpr 4127  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-con 21025 This theorem is referenced by:  iscon2  21027  conclo  21028  conndisj  21029  contop  21030
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