Theorem inf1

Description: Variation of Axiom of Infinity (using zfinf as a hypothesis). Axiom of Infinity in FreydScedrov p. 283. (Contributed by NM, 14-Oct-1996) (Revised by David Abernethy, 1-Oct-2013)

	Ref	Expression
Hypothesis	inf1.1	⊢ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) )
Assertion	inf1	⊢ ∃ 𝑥 ( 𝑥 ≠ ∅ ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		inf1.1	⊢ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) )
2		ne0i	⊢ ( 𝑦 ∈ 𝑥 → 𝑥 ≠ ∅ )
3	2	anim1i	⊢ ( ( 𝑦 ∈ 𝑥 ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) → ( 𝑥 ≠ ∅ ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) ) )
4	1 3	eximii	⊢ ∃ 𝑥 ( 𝑥 ≠ ∅ ∧ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ∃ 𝑧 ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥 ) ) )