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Metamath Proof Explorer

Theorem neleqtrrd

Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017) (Proof shortened by Wolf Lammen, 13-Nov-2019)

	Ref	Expression
Hypotheses	neleqtrrd.1	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐵 )
	neleqtrrd.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	neleqtrrd	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		neleqtrrd.1	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐵 )
2		neleqtrrd.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3	2	eqcomd	⊢ ( 𝜑 → 𝐵 = 𝐴 )
4	1 3	neleqtrd	⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐴 )