Theorem 0nep0 4194

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 4156	. . 3 ⊢ ∅ ∈ V
2	1	snnz 3737	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2449	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2364  ∅c0 3446  {csn 3618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-nul 4155

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-nul 3447  df-sn 3624

This theorem is referenced by:  0inp0  4195  opthprc  4710  2dom  6859  exmidpw  6964  exmidpw2en  6968  exmidaclem  7268  pw1dom2  7287