Theorem subne0d

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

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subne0d.3	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
Assertion	subne0d	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subne0d.3	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
4		subeq0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
5	1 2 4	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = 0 ↔ 𝐴 = 𝐵 ) )
6	5	necon3bid	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) ≠ 0 ↔ 𝐴 ≠ 𝐵 ) )
7	3 6	mpbird	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ≠ 0 )