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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 5311 | . . 3 ⊢ ∅ ≠ {∅} | |
2 | neeq1 3004 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅})) | |
3 | 1, 2 | mpbiri 257 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅}) |
4 | 3 | neneqd 2946 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ≠ wne 2941 ∅c0 4280 {csn 4584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-v 3445 df-dif 3911 df-nul 4281 df-sn 4585 |
This theorem is referenced by: eqsnuniex 5314 dtruALT 5341 zfpair 5374 |
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