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Mirrors > Home > MPE Home > Th. List > 0inp0 | Structured version Visualization version GIF version |
Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
0inp0 | ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nep0 5363 | . . 3 ⊢ ∅ ≠ {∅} | |
2 | neeq1 3000 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅})) | |
3 | 1, 2 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅}) |
4 | 3 | neneqd 2942 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ≠ wne 2937 ∅c0 4338 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-nul 5311 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-nul 4339 df-sn 4631 |
This theorem is referenced by: eqsnuniex 5366 dtruALT 5393 zfpair 5426 |
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