Theorem unisuc

Description: A transitive class is equal to the union of its successor, inference form. Combines Theorem 4E of Enderton p. 72 and Exercise 6 of Enderton p. 73. (Contributed by NM, 30-Aug-1993)

	Ref	Expression
Hypothesis	unisuc.1	$$⊢A\in \mathrm{V}$$
Assertion	unisuc	$$⊢\mathrm{Tr}A\leftrightarrow \bigcup \mathrm{suc}A=A$$

Proof

Step	Hyp	Ref	Expression
1		unisuc.1	$$⊢A\in \mathrm{V}$$
2		unisucg	$$⊢A\in \mathrm{V}\to \left(\mathrm{Tr}A\leftrightarrow \bigcup \mathrm{suc}A=A\right)$$
3	1 2	ax-mp	$$⊢\mathrm{Tr}A\leftrightarrow \bigcup \mathrm{suc}A=A$$