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Mirrors > Home > MPE Home > Th. List > 0nep0 | Structured version Visualization version GIF version |
Description: The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.) |
Ref | Expression |
---|---|
0nep0 | ⊢ ∅ ≠ {∅} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | snnz 4252 | . 2 ⊢ {∅} ≠ ∅ |
3 | 2 | necomi 2836 | 1 ⊢ ∅ ≠ {∅} |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2780 ∅c0 3874 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-nul 3875 df-sn 4126 |
This theorem is referenced by: 0inp0 4763 opthprc 5089 2dom 7915 pw2eng 7951 hashge3el3dif 13122 isusp 21875 bj-1upln0 32190 clsk1indlem0 37359 |
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