Theorem nd3

Description: A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002)

	Ref	Expression
Assertion	nd3	$$⊢\forall xx=y\to \neg \forall zx\in y$$

Proof

Step	Hyp	Ref	Expression
1		elirrv	$$⊢\neg x\in x$$
2		elequ2	$$⊢x=y\to \left(x\in x\leftrightarrow x\in y\right)$$
3	1 2	mtbii	$$⊢x=y\to \neg x\in y$$
4	3	sps	$$⊢\forall xx=y\to \neg x\in y$$
5		sp	$$⊢\forall zx\in y\to x\in y$$
6	4 5	nsyl	$$⊢\forall xx=y\to \neg \forall zx\in y$$