Theorem nfnel 2477

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 2471	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfnel.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfnel.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfel 2356	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
5	4	nfn 1680	. 2 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵
6	1, 5	nfxfr 1496	1 ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵

Syntax hints:  ¬ wn 3  Ⅎwnf 1482   ∈ wcel 2175  Ⅎwnfc 2334   ∉ wnel 2470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-cleq 2197  df-clel 2200  df-nfc 2336  df-nel 2471

This theorem is referenced by: (None)