Theorem 0inp0 1888

Description: Something cannot be equal to both the null set and the power set of the null set.

Assertion

Ref	Expression
0inp0	⊢ (A = ∅ → ¬ A = {∅})

Proof of Theorem 0inp0

Step	Hyp	Ref	Expression
1		0nep0 1887	. 2 ⊢ ¬ ∅ = {∅}
2		cleq1 1107	. 2 ⊢ (A = ∅ → (A = {∅} ↔ ∅ = {∅}))
3	1, 2	mtbiri 539	1 ⊢ (A = ∅ → ¬ A = {∅})

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091  ∅c0 1707  {csn 1808

This theorem is referenced by:  dtru 1889  zfpair 1891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812

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