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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 5197 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | snnz 4689 | . 2 ⊢ {∅} ≠ ∅ |
3 | 2 | necomi 2993 | 1 ⊢ ∅ ≠ {∅} |
Colors of variables: wff setvar class |
Syntax hints: ≠ wne 2939 ∅c0 4234 {csn 4538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-nul 5196 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2940 df-v 3407 df-dif 3866 df-nul 4235 df-sn 4539 |
This theorem is referenced by: 0inp0 5247 opthprc 5610 2dom 8704 pw2eng 8748 djuexb 9522 hashge3el3dif 14049 cat1 17600 isusp 23156 bj-1upln0 34933 clsk1indlem0 41326 mnuprdlem1 41561 mnuprdlem2 41562 |
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