Theorem nnel 3044

Description: Negation of negated membership, analogous to nne 2936. (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 3036	. . 3 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2	1	bicomi 214	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ 𝐴 ∉ 𝐵)
3	2	con1bii 345	1 ⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 196   ∈ wcel 2139   ∉ wnel 3035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-nel 3036

This theorem is referenced by:  raldifsnb  4471  mpt2xopynvov0g  7509  0mnnnnn0  11517  ssnn0fi  12978  rabssnn0fi  12979  hashnfinnn0  13344  lcmfunsnlem2lem2  15554  finsumvtxdg2ssteplem1  26651  pthdivtx  26835  wwlksnndef  27023  wwlksnfi  27024  frgrwopreglem4a  27464  poimirlem26  33748  lswn0  41890  prminf2  42010