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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 5358 | . . 3 ⊢ ∅ ≠ {∅} | |
| 2 | neeq1 3003 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅})) | |
| 3 | 1, 2 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅}) |
| 4 | 3 | neneqd 2945 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ≠ wne 2940 ∅c0 4333 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-nul 4334 df-sn 4627 |
| This theorem is referenced by: eqsnuniex 5361 dtruALT 5388 zfpair 5421 |
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