Theorem 0nep0 5330

Description: The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)

Assertion

Ref	Expression
0nep0	⊢ ∅ ≠ {∅}

Proof of Theorem 0nep0

Step	Hyp	Ref	Expression
1		0ex 5272	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4744	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3014	1 ⊢ ∅ ≠ {∅}

This proof depends on syntax axioms:   ≠ wne 2960  ∅c0 4286  {csn 4591

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 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-nul 4287  df-sn 4592

This theorem is used by:  0inp0  5331  opthprc  5727  2dom  9034  pw2eng  9078  djuexb  9911  hashge3el3dif  14542  cat1  18176  isusp  24469  kard0b  35629  bj-1upln0  37702  clsk1indlem0  44825  mnuprdlem1  45040  mnuprdlem2  45041