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| Mirrors > Home > MPE Home > Th. List > nel0 | Structured version Visualization version GIF version | ||
| Description: From the general negation of membership in 𝐴, infer that 𝐴 is the empty set. (Contributed by BJ, 6-Oct-2018.) | 
| Ref | Expression | 
|---|---|
| nel0.1 | ⊢ ¬ 𝑥 ∈ 𝐴 | 
| Ref | Expression | 
|---|---|
| nel0 | ⊢ 𝐴 = ∅ | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eq0 4349 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
| 2 | nel0.1 | . 2 ⊢ ¬ 𝑥 ∈ 𝐴 | |
| 3 | 1, 2 | mpgbir 1798 | 1 ⊢ 𝐴 = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 = wceq 1539 ∈ wcel 2107 ∅c0 4332 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-dif 3953 df-nul 4333 | 
| This theorem is referenced by: iun0 5061 0iun 5062 0xp 5783 dm0 5930 cnv0 6159 fzouzdisj 13736 bj-ccinftydisj 37215 finxp0 37393 stoweidlem44 46064 | 
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