Theorem 0inp0 5359

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

Assertion

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

Proof of Theorem 0inp0

Step	Hyp	Ref	Expression
1		0nep0 5358	. . 3 ⊢ ∅ ≠ {∅}
2		neeq1 3003	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
3	1, 2	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅})
4	3	neneqd 2945	1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ≠ wne 2940  ∅c0 4333  {csn 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-nul 4334  df-sn 4627

This theorem is referenced by:  eqsnuniex  5361  dtruALT  5388  zfpair  5421