Theorem ac6s5

Description: Generalization of the Axiom of Choice to proper classes. B is a collection B ( x ) of nonempty, possible proper classes. Remark after Theorem 10.46 of TakeutiZaring p. 98. (Contributed by NM, 27-Mar-2006)

	Ref	Expression
Hypothesis	ac6s4.1	$$⊢A\in \mathrm{V}$$
Assertion	ac6s5	$$⊢\forall x\in AB\ne \varnothing \to \exists f\forall x\in Af\left(x\right)\in B$$

Proof

Step	Hyp	Ref	Expression
1		ac6s4.1	$$⊢A\in \mathrm{V}$$
2	1	ac6s4	$$⊢\forall x\in AB\ne \varnothing \to \exists f\left(fFnA\wedge \forall x\in Af\left(x\right)\in B\right)$$
3		exsimpr	$$⊢\exists f\left(fFnA\wedge \forall x\in Af\left(x\right)\in B\right)\to \exists f\forall x\in Af\left(x\right)\in B$$
4	2 3	syl	$$⊢\forall x\in AB\ne \varnothing \to \exists f\forall x\in Af\left(x\right)\in B$$