Theorem 0inp0 5327

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 5326	. . 3 ⊢ ∅ ≠ {∅}
2		neeq1 3019	. . 3 ⊢ (𝐴 = ∅ → (𝐴 ≠ {∅} ↔ ∅ ≠ {∅}))
3	1, 2	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → 𝐴 ≠ {∅})
4	3	neneqd 2962	1 ⊢ (𝐴 = ∅ → ¬ 𝐴 = {∅})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ≠ wne 2957  ∅c0 4282  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-nul 4283  df-sn 4588

This theorem is used by:  eqsnuniex  5330  dtruALT  5357  zfpair  5390