Theorem 0inp0 5235

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 5234	. . 3 ⊢ ∅ ≠ {∅}
2		neeq1 2994	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
3	1, 2	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅})
4	3	neneqd 2937	1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅})

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1543   ≠ wne 2932  ∅c0 4223  {csn 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-nul 5184

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-v 3400  df-dif 3856  df-nul 4224  df-sn 4528

This theorem is referenced by:  eqsnuniex  5237  dtruALT  5266  zfpair  5299  dtruALT2  5313