Proof of the shifting index rule for summations (ILIEKMATHPHYSICS)

Proof that a sum can be partitioned into a sum of sums (ILIEKMATHPHYSICS)

c(a1 + a2 + ... + an) = ca1 + ca2 + ... + can (Proof) [ILIEKMATHPHYSICS]

If p | ab, then p | a or p | b where p is prime (Proof) [ILIEKMATHPHYSICS]

a and b are relatively prime iff as + bt = 1 for integers s and t (Proof) [ILIEKMATHPHYSICS]

gcd(a,b) = as + bt for integers s and t, and is the smallest such positive integer (Proof)

If A ∪ C = B ∪ C, A ∩ C = ∅, and B ∩ C = ∅, then A = B (Proof)

Proof of the Commutative Law of Addition (for natural numbers) [ILIEKMATHPHYSICS]

Proof that n + (m++) = (n + m)++ for all natural numbers n and m (ILIEKMATHPHYSICS)

Proof that n + 0 = n for all natural numbers n (ILIEKMATHPHYSICS)

Proof that every subset of a countable set is countable (ILIEKMATHPHYSICS)

If A ⊆ Z+ then A is countable (Proof)

If A is countable and A ~ B, then B is countable (Proof) [ILIEKMATHPHYSICS]

If B ⊆ Z+ is infinite, then there is a bijection f : Z+ → B (Proof) [ILIEKMATHPHYSICS]

Every nonempty finite subset of real numbers has a maximum (Proof) [ILIEKMATHPHYSICS]

|A\B| = |A| - |B| for finite A and B ⊆ A (Proof)

(A\B) ∪ B = A iff B ⊆ A (Proof)

Proof that Z+ is infinite (ILIEKMATHPHYSICS)

If A is finite, then A ~ B iff B is finite and |A| = |B| (Proof) [ILIEKMATHPHYSICS]

If f is injective, then f↾C is injective (Proof) [ILIEKMATHPHYSICS]

Proof that every subset of a finite set is finite (ILIEKMATHPHYSICS)

Proof that |A ∪ B| = |A| + |B| for finite disjoint A and B (ILIEKMATHPHYSICS)

|A| = 1 iff A = {a} for some a (Proof) [ILIEKMATHPHYSICS]

{(a,b)} is a bijection from {a} to {b} Proof [ILIEKMATHPHYSICS]

(a,b) = (c,d) iff a = c and b = d (Proof) - the fundamental property of ordered pairs

If {a,b} = {c,d} and a = c, then b = d (Proof)

{a,b} = {b,a} Proof

Proof of uniqueness of pair sets (for any a and b, {a,b} is unique) [ILIEKMATHPHYSICS]

Proof of uniqueness of singleton sets (for any object a, {a} is unique) [ILIEKMATHPHYSICS]

|A| = 0 iff A = ∅ (Proof) [ILIEKMATHPHYSICS]