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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 4344 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | nel0.1 | . 2 ⊢ ¬ 𝑥 ∈ 𝐴 | |
3 | 1, 2 | mpgbir 1802 | 1 ⊢ 𝐴 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-dif 3952 df-nul 4324 |
This theorem is referenced by: iun0 5066 0iun 5067 0xp 5775 dm0 5921 cnv0 6141 fzouzdisj 13668 bj-ccinftydisj 36094 finxp0 36272 stoweidlem44 44760 |
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