Theorem 0nep0 4253

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 4214	. . 3 ⊢ ∅ ∈ V
2	1	snnz 3789	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 2485	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 2400  ∅c0 3492  {csn 3667

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-nul 4213

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2802  df-dif 3200  df-nul 3493  df-sn 3673

This theorem is referenced by:  0inp0  4254  opthprc  4775  2dom  6975  exmidpw  7093  exmidpw2en  7097  exmidaclem  7413  pw1dom2  7435