Theorem 0inp0 5332

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 5331	. . 3 ⊢ ∅ ≠ {∅}
2		neeq1 3026	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
3	1, 2	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅})
4	3	neneqd 2969	1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ≠ wne 2964  ∅c0 4294  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5273

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-v 3465  df-dif 3916  df-nul 4295  df-sn 4595

This theorem is referenced by:  eqsnuniex  5335  dtruALT  5362  zfpair  5395