Theorem 0nep0 4255

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 4216	. . 3 ⊢ ∅ ∈ V
2	1	snnz 3791	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2487	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2402  ∅c0 3494  {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-nul 4215

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-nul 3495  df-sn 3675

This theorem is referenced by:  0inp0  4256  opthprc  4777  2dom  6979  exmidpw  7099  exmidpw2en  7103  exmidaclem  7422  pw1dom2  7444