Theorem axregndlem1

Description: Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 3-Jan-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	axregndlem1	$$⊢\forall xx=z\to \left(x\in y\to \exists x\left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.8a	$$⊢x\in y\to \exists xx\in y$$
2		nfae	$$⊢Ⅎx\forall xx=z$$
3		nfae	$$⊢Ⅎz\forall xx=z$$
4		elirrv	$$⊢\neg x\in x$$
5		elequ1	$$⊢x=z\to \left(x\in x\leftrightarrow z\in x\right)$$
6	4 5	mtbii	$$⊢x=z\to \neg z\in x$$
7	6	sps	$$⊢\forall xx=z\to \neg z\in x$$
8	7	pm2.21d	$$⊢\forall xx=z\to \left(z\in x\to \neg z\in y\right)$$
9	3 8	alrimi	$$⊢\forall xx=z\to \forall z\left(z\in x\to \neg z\in y\right)$$
10	9	anim2i	$$⊢\left(x\in y\wedge \forall xx=z\right)\to \left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)$$
11	10	expcom	$$⊢\forall xx=z\to \left(x\in y\to \left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)\right)$$
12	2 11	eximd	$$⊢\forall xx=z\to \left(\exists xx\in y\to \exists x\left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)\right)$$
13	1 12	syl5	$$⊢\forall xx=z\to \left(x\in y\to \exists x\left(x\in y\wedge \forall z\left(z\in x\to \neg z\in y\right)\right)\right)$$