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Metamath Proof Explorer

Theorem eldifbd

Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif . (Contributed by David Moews, 1-May-2017)

		Ref	Expression
	Hypothesis	eldifbd.1	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) )
	Assertion	eldifbd	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eldifbd.1	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) )
2		eldif	⊢ ( 𝐴 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) )
3	1 2	sylib	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶 ) )
4	3	simprd	⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐶 )