Theorem elnanel

Description: Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp and serves as an example in the context of Godel codes, see elnanelprv . (Contributed by AV, 5-Nov-2023) (New usage is discouraged.)

	Ref	Expression
Assertion	elnanel	$$⊢\left(A\in B⊼B\in A\right)$$

Proof

Step	Hyp	Ref	Expression
1		en2lp	$$⊢\neg \left(A\in B\wedge B\in A\right)$$
2		df-nan	$$⊢\left(A\in B⊼B\in A\right)\leftrightarrow \neg \left(A\in B\wedge B\in A\right)$$
3	1 2	mpbir	$$⊢\left(A\in B⊼B\in A\right)$$