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Metamath Proof Explorer

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 ∈ V
	Assertion	unisuc	⊢ ( Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		unisuc.1	⊢ 𝐴 ∈ V
2		unisucg	⊢ ( 𝐴 ∈ V → ( Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴 ) )
3	1 2	ax-mp	⊢ ( Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴 )