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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	$⊢ φ \to \neg C \in B$
	neleqtrrd.2	$⊢ φ \to A = B$
Assertion	neleqtrrd	$⊢ φ \to \neg C \in A$

Proof

Step	Hyp	Ref	Expression
1		neleqtrrd.1	$⊢ φ \to \neg C \in B$
2		neleqtrrd.2	$⊢ φ \to A = B$
3	2	eqcomd	$⊢ φ \to B = A$
4	1 3	neleqtrd	$⊢ φ \to \neg C \in A$