ordsuci

Description: The successor of an ordinal class is an ordinal class. Remark 1.5 of Schloeder p. 1. (Contributed by NM, 6-Jun-1994) Extract and adapt from a subproof of onsuc . (Revised by BTernaryTau, 6-Jan-2025) (Proof shortened by BJ, 11-Jan-2025)

		Ref	Expression
	Assertion	ordsuci	$⊢ Ord A \to Ord suc A$

Proof

Step	Hyp	Ref	Expression
1		ordtr	$⊢ Ord A \to Tr A$
2		suctr	$⊢ Tr A \to Tr suc A$
3	1 2	syl	$⊢ Ord A \to Tr suc A$
4		df-suc	$⊢ suc A = A \cup \{A\}$
5		ordsson	$⊢ Ord A \to A \subseteq On$
6		elon2	$⊢ A \in On \leftrightarrow (Ord A \land A \in V)$
7		snssi	$⊢ A \in On \to \{A\} \subseteq On$
8	6 7	sylbir	$⊢ (Ord A \land A \in V) \to \{A\} \subseteq On$
9		snprc	$⊢ \neg A \in V \leftrightarrow \{A\} = \emptyset$
10	9	biimpi	$⊢ \neg A \in V \to \{A\} = \emptyset$
11		0ss	$⊢ \emptyset \subseteq On$
12	10 11	eqsstrdi	$⊢ \neg A \in V \to \{A\} \subseteq On$
13	12	adantl	$⊢ (Ord A \land \neg A \in V) \to \{A\} \subseteq On$
14	8 13	pm2.61dan	$⊢ Ord A \to \{A\} \subseteq On$
15	5 14	unssd	$⊢ Ord A \to A \cup \{A\} \subseteq On$
16	4 15	eqsstrid	$⊢ Ord A \to suc A \subseteq On$
17		ordon	$⊢ Ord On$
18	17	a1i	$⊢ Ord A \to Ord On$
19		trssord	$⊢ (Tr suc A \land suc A \subseteq On \land Ord On) \to Ord suc A$
20	3 16 18 19	syl3anc	$⊢ Ord A \to Ord suc A$

Metamath Proof Explorer

Theorem ordsuci

Proof