http://us2.metamath.org:8888/mpeuni/mmrecent.html Recent proofs for Metamath proof system en http://us2.metamath.org:8888/mpeuni/comptoppr.html 3-Jul-2009   ⊢ ((J ∈ Top ⋀ K ∈ Top ⋀ J ~= K) → (J ∈ Comp ↔ K ∈ Comp))
    ]]> http://us2.metamath.org:8888/mpeuni/comptopprlem.html 3-Jul-2009comptoppr 11089.
   ⊢ ((J ∈ Top ⋀ K ∈ Top ⋀ J ~= K) → (J ∈ Comp → K ∈ Comp))
    ]]> http://us2.metamath.org:8888/mpeuni/com15.html 3-Jul-2009   ⊢ (φ → (ψ → (χ → (θ → (τ → η)))))    ⇒
   ⊢ (τ → (ψ → (χ → (θ → (φ → η)))))
    ]]> http://us2.metamath.org:8888/mpeuni/com45.html 3-Jul-2009   ⊢ (φ → (ψ → (χ → (θ → (τ → η)))))    ⇒
   ⊢ (φ → (ψ → (χ → (τ → (θ → η)))))
    ]]> http://us2.metamath.org:8888/mpeuni/cj11.html 2-Jul-2009cj11OLD 7027.)
   ⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → ((∗ ‘A) = (∗ ‘B) ↔ A = B))
    ]]> http://us2.metamath.org:8888/mpeuni/laspwcl.html 1-Jul-2009   ⊢ X = dom R    ⇒
   ⊢ ((R ∈ Lat ⋀ A ∈ X ⋀ B ∈ X) → (R sup_w {A, B}) ∈ X)
    ]]> http://us2.metamath.org:8888/mpeuni/cptclsscpt.html 30-Jun-2009   ⊢ ((J ∈ Comp ⋀ S ∈ (Clsd ‘J)) → (subSp ‘⟨S, J⟩) ∈ Comp)
    ]]> http://us2.metamath.org:8888/mpeuni/compsub.html 30-Jun-2009Beginning Topology.
   ⊢ X = ∪J    ⇒
   ⊢ ((J ∈ Top ⋀ S ⊆ X) → ((subSp ‘⟨S, J⟩) ∈ Comp ↔ ∀c ∈ ℘ J(S ⊆ ∪c → ∃d ∈ (℘c ∩ Fin)S ⊆ ∪d)))
    ]]> http://us2.metamath.org:8888/mpeuni/compsublem.html 30-Jun-2009compsub 11084.
   ⊢ X = ∪J    ⇒
   ⊢ ((J ∈ Top ⋀ S ⊆ X) → (∀c ∈ ℘ J(S ⊆ ∪c → ∃d ∈ (℘c ∩ Fin)S ⊆ ∪d) → ∀s ∈ ℘ (subSp ‘⟨S, J⟩)(∪(subSp ‘⟨S, J⟩) = ∪s → ∃t ∈ (℘s ∩ Fin)∪(subSp ‘⟨S, J⟩) = ∪t)))
    ]]> http://us2.metamath.org:8888/mpeuni/compcov.html 30-Jun-2009   ⊢ X = ∪J    ⇒
   ⊢ ((J ∈ Comp ⋀ S ⊆ J ⋀ X = ∪S) → ∃s ∈ (℘S ∩ Fin)X = ∪s)]]>