Theorem elhf

Description: Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015)

	Ref	Expression
Assertion	elhf	⊢ ( 𝐴 ∈ Hf ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		df-hf	⊢ Hf = ∪ ( 𝑅1 “ ω )
2	1	eleq2i	⊢ ( 𝐴 ∈ Hf ↔ 𝐴 ∈ ∪ ( 𝑅1 “ ω ) )
3		r111	⊢ 𝑅1 : On –1-1→ V
4		f1fun	⊢ ( 𝑅1 : On –1-1→ V → Fun 𝑅1 )
5		eluniima	⊢ ( Fun 𝑅1 → ( 𝐴 ∈ ∪ ( 𝑅1 “ ω ) ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) ) )
6	3 4 5	mp2b	⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ ω ) ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )
7	2 6	bitri	⊢ ( 𝐴 ∈ Hf ↔ ∃ 𝑥 ∈ ω 𝐴 ∈ ( 𝑅1 ‘ 𝑥 ) )