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Metamath Proof Explorer

Theorem dfnul2

Description: Alternate definition of the empty set. Definition 5.14 of TakeutiZaring p. 20. (Contributed by NM, 26-Dec-1996) Remove dependency on ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 3-May-2023) (Proof shortened by BJ, 23-Sep-2024)

		Ref	Expression
	Assertion	dfnul2	$⊢ \emptyset = \{x \| \neg x = x\}$

Proof

Step	Hyp	Ref	Expression
1		dfnul4	$⊢ \emptyset = \{x \| ⊥\}$
2		equid	$⊢ x = x$
3	2	notnoti	$⊢ \neg \neg x = x$
4	3	bifal	$⊢ \neg x = x \leftrightarrow ⊥$
5	4	abbii	$⊢ \{x \| \neg x = x\} = \{x \| ⊥\}$
6	1 5	eqtr4i	$⊢ \emptyset = \{x \| \neg x = x\}$