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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 1793 | 1 ⊢ 𝐴 = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-dif 3947 df-nul 4323 |
This theorem is referenced by: iun0 5066 0iun 5067 0xp 5776 dm0 5923 cnv0 6147 fzouzdisj 13703 bj-ccinftydisj 36823 finxp0 37001 stoweidlem44 45570 |
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