Theorem eq0 1719

Description: The empty set has no elements. Theorem 2 of [Suppes] p. 22.

Assertion

Ref	Expression
eq0	⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)

Distinct variable group(s):   x,A

Proof of Theorem eq0

Step	Hyp	Ref	Expression
1		n0 1714	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
2		df-ex 679	. . 3 ⊢ (∃x x ∈ A ↔ ¬ ∀x ¬ x ∈ A)
3	1, 2	bitr 151	. 2 ⊢ (¬ A = ∅ ↔ ¬ ∀x ¬ x ∈ A)
4	3	bicon4i 401	1 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)

Syntax hints:  ¬ wn 1   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∅c0 1707

This theorem is referenced by:  0el 1720  ssdif0 1748  difin0ss 1753  inssdif0 1754  reldm0 2550  tz6.12-2 2845  uzwo 4605  nnwoOLD 4608

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-16 922  ax-17 925  ax-ext 1074

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-nul 1708

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