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| Mirrors > Home > ILE Home > Th. List > nel0 | 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 3479 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
| 2 | nel0.1 | . 2 ⊢ ¬ 𝑥 ∈ 𝐴 | |
| 3 | 1, 2 | mpgbir 1476 | 1 ⊢ 𝐴 = ∅ |
| Colors of variables: wff set class |
| Syntax hints: ¬ wn 3 = wceq 1373 ∈ wcel 2176 ∅c0 3460 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-v 2774 df-dif 3168 df-nul 3461 |
| This theorem is referenced by: (None) |
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