Theorem nfnel 2438

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

Hypotheses

Ref	Expression
nfnel.1	⊢ Ⅎ𝑥𝐴
nfnel.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfnel	⊢ Ⅎ𝑥 𝐴 ∉ 𝐵

Proof of Theorem nfnel

Step	Hyp	Ref	Expression
1		df-nel 2432	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfnel.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfnel.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfel 2317	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
5	4	nfn 1646	. 2 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵
6	1, 5	nfxfr 1462	1 ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵

Syntax hints:  ¬ wn 3  Ⅎwnf 1448   ∈ wcel 2136  Ⅎwnfc 2295   ∉ wnel 2431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-nel 2432

This theorem is referenced by: (None)