Theorem ac4c

Description: Equivalent of Axiom of Choice (class version). (Contributed by NM, 10-Feb-1997)

	Ref	Expression
Hypothesis	ac4c.1	$$⊢A\in \mathrm{V}$$
Assertion	ac4c	$$⊢\exists f\forall x\in A\left(x\ne \varnothing \to f\left(x\right)\in x\right)$$

Proof

Step	Hyp	Ref	Expression
1		ac4c.1	$$⊢A\in \mathrm{V}$$
2		raleq	$$⊢y=A\to \left(\forall x\in y\left(x\ne \varnothing \to f\left(x\right)\in x\right)\leftrightarrow \forall x\in A\left(x\ne \varnothing \to f\left(x\right)\in x\right)\right)$$
3	2	exbidv	$$⊢y=A\to \left(\exists f\forall x\in y\left(x\ne \varnothing \to f\left(x\right)\in x\right)\leftrightarrow \exists f\forall x\in A\left(x\ne \varnothing \to f\left(x\right)\in x\right)\right)$$
4		ac4	$$⊢\exists f\forall x\in y\left(x\ne \varnothing \to f\left(x\right)\in x\right)$$
5	1 3 4	vtocl	$$⊢\exists f\forall x\in A\left(x\ne \varnothing \to f\left(x\right)\in x\right)$$