Given
∀x∀y(xRy ∧ x≠y => ¬yRx)
Prove that
∀x∀y(xRy ∧ yRx => x=y)
Proof:

  Suppose a and b are arbitrary objects, it follows (by definition)
aRb ∧ a≠b => ¬bRa
  Suppose: aRb ∧ bRa
  To be proven: a=b

  Proof:
    Suppose: a≠b
    To be proven: ⊥ (ie contradiction)

    Proof:
      From aRb ∧ a≠b => ¬bRa, and aRb ∧ bRa,
      it follows ¬bRa ∧ bRa ≡ ⊥

    Thus a=b

  Thus ∀x∀y(xRy ∧ yRx => x=y)

# posted by rot13(Unafba Pune) @ 8:55 AM