Theorem nnel 3039

Description: Negation of negated membership, analogous to nne 2929. (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 3030	. . 3 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2	1	bicomi 224	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 ↔ 𝐴 ∉ 𝐵)
3	2	con1bii 356	1 ⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∈ wcel 2109   ∉ wnel 3029

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 207  df-nel 3030

This theorem is referenced by:  raldifsnb  4747  mpoxopynvov0g  8147  fsetexb  8791  0mnnnnn0  12416  ssnn0fi  13892  rabssnn0fi  13893  hashnfinnn0  14268  lcmfunsnlem2lem2  16550  finsumvtxdg2ssteplem1  29491  pthdivtx  29672  wwlksnndef  29850  frgrwopreglem4a  30254  poimirlem26  37636  sticksstones1  42129  afv2orxorb  47222  afv2fv0  47259  lswn0  47438  prminf2  47582