http://us2.metamath.org:8888/mpeuni/mmrecent.html Recent proofs for Metamath proof system en http://us2.metamath.org:8888/mpeuni/rpneg.html 20-Nov-2008Apostol] p. 20.
   ⊢ ((A ∈ ℝ ⋀ A ≠ 0) → (A ∈ ℝ⁺ ↔ ¬ -A ∈ ℝ⁺))
    ]]> http://us2.metamath.org:8888/mpeuni/pwne.html 19-Nov-2008   ⊢ (A ∈ B → ℘A ≠ A)
    ]]> http://us2.metamath.org:8888/mpeuni/rerpdivcl.html 18-Nov-2008   ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ⁺) → (A / B) ∈ ℝ)
    ]]> http://us2.metamath.org:8888/mpeuni/sbc8g.html 18-Nov-2008df-sbc 1949 if we start from dfsbcq 1950 (see its comments).
   ⊢ (A ∈ B → ([A / x]φ ↔ A ∈ {x∣φ}))
    ]]> http://us2.metamath.org:8888/mpeuni/sbc7g.html 18-Nov-2008df-clab 1470. Note that x and A don't have to be distinct.
   ⊢ (A ∈ B → ([A / x]φ ↔ ∃y(y = A ⋀ [y / x]φ)))
    ]]> http://us2.metamath.org:8888/mpeuni/isepic.html 17-Nov-2008F belongs to a homset.
   ⊢ O = dom (id ‘T)    &
   ⊢ H = (hom ‘T)    &
   ⊢ R = (o ‘T)    ⇒
   ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O) ⋀ F ∈ (H ‘⟨A, B⟩)) → (F ∈ (Epi ‘T) ↔ ∀c ∈ O ∀g ∈ (H ‘⟨B, c⟩)∀h ∈ (H ‘⟨B, c⟩)((gRF) = (hRF) → g = h)))
    ]]> http://us2.metamath.org:8888/mpeuni/usinuniop.html 17-Nov-2008   ⊢ X = ∪J    ⇒
   ⊢ (J ∈ Con → ∀x ∈ J ∀y ∈ J ((x ≠ ∅ ⋀ y ≠ ∅ ⋀ (x ∩ y) = ∅) → X ≠ (x ∪ y)))
    ]]> http://us2.metamath.org:8888/mpeuni/iscon2.html 17-Nov-2008J is a connected topology .
   ⊢ (J ∈ Con ↔ (J ∈ Top ⋀ (J ∩ (Clsd ‘J)) = {∅, ∪J}))
    ]]> http://us2.metamath.org:8888/mpeuni/iscon.html 17-Nov-2008J is a connected topology .
   ⊢ (J ∈ Top → (J ∈ Con ↔ (J ∩ (Clsd ‘J)) = {∅, ∪J}))
    ]]> http://us2.metamath.org:8888/mpeuni/vri.html 17-Nov-2008W was chosen because V is already used for the universal class.
   ⊢ G = (1^st ‘W)    &
   ⊢ S = (2^nd ‘W)    &
   ⊢ X = ran G    ⇒
   ⊢ (W ∈ RVec → (G ∈ Abel ⋀ S:(ℝ × X)–→X ⋀ ∀x ∈ X ((1Sx) = x ⋀ ∀y ∈ ℝ (∀z ∈ X (yS(xGz)) = ((ySx)G(ySz)) ⋀ ∀z ∈ ℝ (((y + z)Sx) = ((ySx)G(zSx)) ⋀ ((y · z)Sx) = (yS(zSx)))))))]]>