Theorem 0nep0 4199

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 4161	. . 3 ⊢ ∅ ∈ V
2	1	snnz 3742	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2452	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2367  ∅c0 3451  {csn 3623

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-nul 4160

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-nul 3452  df-sn 3629

This theorem is referenced by:  0inp0  4200  opthprc  4715  2dom  6873  exmidpw  6978  exmidpw2en  6982  exmidaclem  7291  pw1dom2  7310