Theorem 0nep0 5249

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 5202	. . 3 ⊢ ∅ ∈ V
2	1	snnz 4703	. 2 ⊢ {∅} ≠ ∅
3	2	necomi 3068	1 ⊢ ∅ ≠ {∅}

Syntax hints:   ≠ wne 3014  ∅c0 4289  {csn 4559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-v 3495  df-dif 3937  df-nul 4290  df-sn 4560

This theorem is referenced by:  0inp0  5250  opthprc  5609  2dom  8574  pw2eng  8615  djuexb  9330  hashge3el3dif  13836  isusp  22862  bj-1upln0  34314  clsk1indlem0  40382  mnuprdlem1  40599  mnuprdlem2  40600