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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 4343 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | nel0.1 | . 2 ⊢ ¬ 𝑥 ∈ 𝐴 | |
3 | 1, 2 | mpgbir 1801 | 1 ⊢ 𝐴 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-dif 3951 df-nul 4323 |
This theorem is referenced by: iun0 5065 0iun 5066 0xp 5774 dm0 5920 cnv0 6140 fzouzdisj 13667 bj-ccinftydisj 36089 finxp0 36267 stoweidlem44 44750 |
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