Theorem 0inp0 5329

Description: Something cannot be equal to both the empty set and the power set of the empty set. (Contributed by NM, 21-Jun-1993.)

Assertion

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

Proof of Theorem 0inp0

Step	Hyp	Ref	Expression
1		0nep0 5328	. . 3 ⊢ ∅ ≠ {∅}
2		neeq1 3020	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
3	1, 2	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅})
4	3	neneqd 2963	1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ≠ wne 2958  ∅c0 4286  {csn 4589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-nul 4287  df-sn 4590

This theorem is used by:  eqsnuniex  5332  dtruALT  5359  zfpair  5392