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Metamath Proof Explorer

Theorem pred0

Description: The predecessor class over (/) is always (/) . (Contributed by Scott Fenton, 16-Apr-2011) (Proof shortened by AV, 11-Jun-2021)

		Ref	Expression
	Assertion	pred0	$⊢ Pred (R, \emptyset, X) = \emptyset$

Step	Hyp	Ref	Expression
1		df-pred	$⊢ Pred (R, \emptyset, X) = \emptyset \cap R^{-1} [\{X\}]$
2		0in	$⊢ \emptyset \cap R^{-1} [\{X\}] = \emptyset$
3	1 2	eqtri	$⊢ Pred (R, \emptyset, X) = \emptyset$