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Mirrors > Home > MPE Home > Th. List > nnel | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nnel | ⊢ (¬ 𝐴 ∉ 𝐵 ↔ 𝐴 ∈ 𝐵) |
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) |
Colors of variables: wff setvar class |
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 |
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