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Mirrors > Home > ILE Home > Th. List > 0nep0 | 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 4025 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | snnz 3612 | . 2 ⊢ {∅} ≠ ∅ |
3 | 2 | necomi 2370 | 1 ⊢ ∅ ≠ {∅} |
Colors of variables: wff set class |
Syntax hints: ≠ wne 2285 ∅c0 3333 {csn 3497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-v 2662 df-dif 3043 df-nul 3334 df-sn 3503 |
This theorem is referenced by: 0inp0 4060 opthprc 4560 2dom 6667 exmidpw 6770 exmidaclem 7032 pw1dom2 13117 |
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