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	$$⊢\exists x\left(y\in x\wedge \forall y\left(y\in x\to \exists z\left(y\in z\wedge z\in x\right)\right)\right)$$
Assertion	inf1	$$⊢\exists x\left(x\ne \varnothing \wedge \forall y\left(y\in x\to \exists z\left(y\in z\wedge z\in x\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		inf1.1	$$⊢\exists x\left(y\in x\wedge \forall y\left(y\in x\to \exists z\left(y\in z\wedge z\in x\right)\right)\right)$$
2		ne0i	$$⊢y\in x\to x\ne \varnothing$$
3	2	anim1i	$$⊢\left(y\in x\wedge \forall y\left(y\in x\to \exists z\left(y\in z\wedge z\in x\right)\right)\right)\to \left(x\ne \varnothing \wedge \forall y\left(y\in x\to \exists z\left(y\in z\wedge z\in x\right)\right)\right)$$
4	1 3	eximii	$$⊢\exists x\left(x\ne \varnothing \wedge \forall y\left(y\in x\to \exists z\left(y\in z\wedge z\in x\right)\right)\right)$$