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Theorem 0inp0 3910
 Description: Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)
Assertion
Ref Expression
0inp0 (A = ∅ → ¬ A = {∅})

Proof of Theorem 0inp0
StepHypRef Expression
1 0nep0 3909 . . 3 ∅ ≠ {∅}
2 neeq1 2213 . . 3 (A = ∅ → (A ≠ {∅} ↔ ∅ ≠ {∅}))
31, 2mpbiri 157 . 2 (A = ∅ → A ≠ {∅})
43neneqd 2221 1 (A = ∅ → ¬ A = {∅})
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1242   ≠ wne 2201  ∅c0 3218  {csn 3367 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-nul 3219  df-sn 3373 This theorem is referenced by: (None)
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