Theorem ac6s4

Description: Generalization of the Axiom of Choice to proper classes. B is a collection B ( x ) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006)

	Ref	Expression
Hypothesis	ac6s4.1	$$⊢A\in \mathrm{V}$$
Assertion	ac6s4	$$⊢\forall x\in AB\ne \varnothing \to \exists f\left(fFnA\wedge \forall x\in Af\left(x\right)\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ac6s4.1	$$⊢A\in \mathrm{V}$$
2		n0	$$⊢B\ne \varnothing \leftrightarrow \exists yy\in B$$
3	2	ralbii	$$⊢\forall x\in AB\ne \varnothing \leftrightarrow \forall x\in A\exists yy\in B$$
4		eleq1	$$⊢y=f\left(x\right)\to \left(y\in B\leftrightarrow f\left(x\right)\in B\right)$$
5	1 4	ac6s2	$$⊢\forall x\in A\exists yy\in B\to \exists f\left(fFnA\wedge \forall x\in Af\left(x\right)\in B\right)$$
6	3 5	sylbi	$$⊢\forall x\in AB\ne \varnothing \to \exists f\left(fFnA\wedge \forall x\in Af\left(x\right)\in B\right)$$