Theorem 0nep0 4261

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 4221	. . 3 ⊢ ∅ ∈ V
2	1	snnz 3795	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2488	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2403  ∅c0 3496  {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-nul 3497  df-sn 3679

This theorem is referenced by:  0inp0  4262  opthprc  4783  2dom  7023  exmidpw  7143  exmidpw2en  7147  exmidaclem  7466  pw1dom2  7488