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Mirrors > Home > ILE Home > Th. List > 0inp0 | GIF version |
Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.) |
Ref | Expression |
---|---|
0inp0 | ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nep0 4126 | . . 3 ⊢ ∅ ≠ {∅} | |
2 | neeq1 2340 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅})) | |
3 | 1, 2 | mpbiri 167 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅}) |
4 | 3 | neneqd 2348 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1335 ≠ wne 2327 ∅c0 3394 {csn 3560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 ax-nul 4090 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-v 2714 df-dif 3104 df-nul 3395 df-sn 3566 |
This theorem is referenced by: (None) |
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