if the first integer is greater than the second integer then the opposite of the first integer is less than the opposite of the second integer?

Yes, if the first integer (let's call it ( a )) is greater than the second integer (let's call it ( b )), then the opposite (or negative) of the first integer (-( a )) will indeed be less than the opposite of the second integer (-( b )). This is because negating both integers reverses their order; since ( a > b ), it follows that -( a < -b ).