Theorem axregprim

Description: ax-reg without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010)

	Ref	Expression
Assertion	axregprim	$$⊢x\in y\to \neg \forall x\left(x\in y\to \neg \forall z\left(z\in x\to \neg z\in y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		axregnd	$$⊢x\in y\to \exists x\left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)$$
2		df-an	$$⊢\left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)\leftrightarrow \neg \left(x\in y\to \neg \forall z\left(z\in x\to \neg z\in y\right)\right)$$
3	2	exbii	$$⊢\exists x\left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)\leftrightarrow \exists x\neg \left(x\in y\to \neg \forall z\left(z\in x\to \neg z\in y\right)\right)$$
4		exnal	$$⊢\exists x\neg \left(x\in y\to \neg \forall z\left(z\in x\to \neg z\in y\right)\right)\leftrightarrow \neg \forall x\left(x\in y\to \neg \forall z\left(z\in x\to \neg z\in y\right)\right)$$
5	3 4	bitri	$$⊢\exists x\left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)\leftrightarrow \neg \forall x\left(x\in y\to \neg \forall z\left(z\in x\to \neg z\in y\right)\right)$$
6	1 5	sylib	$$⊢x\in y\to \neg \forall x\left(x\in y\to \neg \forall z\left(z\in x\to \neg z\in y\right)\right)$$