sucprcreg

Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004) (Proof shortened by BJ, 16-Apr-2019) (Proof shortened by SN, 22-Apr-2026)

		Ref	Expression
	Assertion	sucprcreg	$⊢ \neg A \in V \leftrightarrow suc A = A$

Proof

Step	Hyp	Ref	Expression
1		sucprc	$⊢ \neg A \in V \to suc A = A$
2		elirr	$⊢ \neg A \in A$
3		snssg	$⊢ A \in V \to (A \in A \leftrightarrow \{A\} \subseteq A)$
4	2 3	mtbii	$⊢ A \in V \to \neg \{A\} \subseteq A$
5		df-suc	$⊢ suc A = A \cup \{A\}$
6	5	eqeq1i	$⊢ suc A = A \leftrightarrow A \cup \{A\} = A$
7		ssequn2	$⊢ \{A\} \subseteq A \leftrightarrow A \cup \{A\} = A$
8	6 7	sylbb2	$⊢ suc A = A \to \{A\} \subseteq A$
9	4 8	nsyl3	$⊢ suc A = A \to \neg A \in V$
10	1 9	impbii	$⊢ \neg A \in V \leftrightarrow suc A = A$

Metamath Proof Explorer

Theorem sucprcreg

Proof