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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 4196 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | nel0.1 | . 2 ⊢ ¬ 𝑥 ∈ 𝐴 | |
3 | 1, 2 | mpgbir 1762 | 1 ⊢ 𝐴 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1507 ∈ wcel 2050 ∅c0 4180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-dif 3834 df-nul 4181 |
This theorem is referenced by: iun0 4852 0iun 4853 0xp 5500 dm0 5638 cnv0 5841 fzouzdisj 12891 bj-ccinftydisj 33964 finxp0 34113 stoweidlem44 41761 |
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