Theorem 0nep0 4217

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 4179	. . 3 ⊢ ∅ ∈ V
2	1	snnz 3757	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2462	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2377  ∅c0 3464  {csn 3638

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-nul 4178

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-v 2775  df-dif 3172  df-nul 3465  df-sn 3644

This theorem is referenced by:  0inp0  4218  opthprc  4734  2dom  6911  exmidpw  7020  exmidpw2en  7024  exmidaclem  7336  pw1dom2  7358