Theorem 0nep0 1887

Description: The empty set and its power set are not equal.

Assertion

Ref	Expression
0nep0	⊢ ¬ ∅ = {∅}

Proof of Theorem 0nep0

Step	Hyp	Ref	Expression
1		0ex 1745	. . 3 ⊢ ∅ ∈ V
2	1	snnz 1846	. 2 ⊢ ¬ {∅} = ∅
3		cleqcom 1103	. 2 ⊢ ({∅} = ∅ ↔ ∅ = {∅})
4	2, 3	mtbi 166	1 ⊢ ¬ ∅ = {∅}

Syntax hints:  ¬ wn 1   = wceq 1091  ∅c0 1707  {csn 1808

This theorem is referenced by:  0inp0 1888  opthprc 2457  0ne1oOLD 3113  map0 3268  2dom 3332  endisj 3341

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812

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