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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 4356 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | nel0.1 | . 2 ⊢ ¬ 𝑥 ∈ 𝐴 | |
3 | 1, 2 | mpgbir 1796 | 1 ⊢ 𝐴 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-dif 3966 df-nul 4340 |
This theorem is referenced by: iun0 5067 0iun 5068 0xp 5787 dm0 5934 cnv0 6163 fzouzdisj 13732 bj-ccinftydisj 37196 finxp0 37374 stoweidlem44 46000 |
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