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Metamath Proof Explorer

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	⊢ ( 𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		en2lp	⊢ ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 )
2		df-nan	⊢ ( ( 𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴 ) ↔ ¬ ( 𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴 ) )
3	1 2	mpbir	⊢ ( 𝐴 ∈ 𝐵 ⊼ 𝐵 ∈ 𝐴 )