Theorem subne0d

Description: Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017)

	Ref	Expression
Hypotheses	negidd.1	|- ( ph -> A e. CC )
	pncand.2	|- ( ph -> B e. CC )
	subne0d.3	|- ( ph -> A =/= B )
Assertion	subne0d	|- ( ph -> ( A - B ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	|- ( ph -> A e. CC )
2		pncand.2	|- ( ph -> B e. CC )
3		subne0d.3	|- ( ph -> A =/= B )
4		subeq0	|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = 0 <-> A = B ) )
5	1 2 4	syl2anc	|- ( ph -> ( ( A - B ) = 0 <-> A = B ) )
6	5	necon3bid	|- ( ph -> ( ( A - B ) =/= 0 <-> A =/= B ) )
7	3 6	mpbird	|- ( ph -> ( A - B ) =/= 0 )