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Metamath Proof Explorer

Theorem nfnel

Description: Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011) (Revised by Mario Carneiro, 7-Oct-2016)

	Ref	Expression
Hypotheses	nfnel.1	⊢ Ⅎ 𝑥 𝐴
	nfnel.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfnel	⊢ Ⅎ 𝑥 𝐴 ∉ 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfnel.1	⊢ Ⅎ 𝑥 𝐴
2		nfnel.2	⊢ Ⅎ 𝑥 𝐵
3		df-nel	⊢ ( 𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵 )
4	1 2	nfel	⊢ Ⅎ 𝑥 𝐴 ∈ 𝐵
5	4	nfn	⊢ Ⅎ 𝑥 ¬ 𝐴 ∈ 𝐵
6	3 5	nfxfr	⊢ Ⅎ 𝑥 𝐴 ∉ 𝐵