Theorem nfnel 3069

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 3062	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		nfnel.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfnel.2	. . . 4 ⊢ Ⅎ𝑥𝐵
4	2, 3	nfel 2936	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
5	4	nfn 1889	. 2 ⊢ Ⅎ𝑥 ¬ 𝐴 ∈ 𝐵
6	1, 5	nfxfr 1885	1 ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵

This proof depends on syntax axioms:  ¬ wn 3  Ⅎwnf 1815   ∈ wcel 2144  Ⅎwnfc 2907   ∉ wnel 3061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-cleq 2752  df-clel 2835  df-nfc 2909  df-nel 3062

This theorem is used by: (None)