Theorem nnel 3071

Description: Negation of negated membership, analogous to nne 2959. (Contributed by Alexander van der Vekens, 18-Jan-2018.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Assertion

Ref	Expression
nnel	⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵)

Proof of Theorem nnel

Step	Hyp	Ref	Expression
1		df-nel 3062	. . 3 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2	1	bicomi 227	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ 𝐴 ∉ 𝐵)
3	2	con1bii 359	1 ⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∈ wcel 2145   ∉ wnel 3061

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-nel 3062

This theorem is used by:  raldifsnb  4758  mpoxopynvov0g  8209  fsetexb  8864  0mnnnnn0  12607  ssnn0fi  14096  rabssnn0fi  14097  hashnfinnn0  14472  lcmfunsnlem2lem2  16776  finsumvtxdg2ssteplem1  30059  pthdivtx  30245  wwlksnndef  30427  frgrwopreglem4a  30844  poimirlem26  38484  sticksstones1  43116  afv2orxorb  48220  afv2fv0  48257  lswn0  48448  nprmmul1  48531  prminf2  48595