Theorem eq0 3565

Description: The empty set has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.)

Assertion

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

Distinct variable group:   x,A

Proof of Theorem eq0

Step	Hyp	Ref	Expression
1		neq0 3561	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
2		df-ex 1542	. . 3 ⊢ (∃x x ∈ A ↔ ¬ ∀x ¬ x ∈ A)
3	1, 2	bitri 240	. 2 ⊢ (¬ A = ∅ ↔ ¬ ∀x ¬ x ∈ A)
4	3	con4bii 288	1 ⊢ (A = ∅ ↔ ∀x ¬ x ∈ A)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  0el  3567  disj  3592  ssdif0  3610  difin0ss  3617  inssdif0  3618  ralf0  3657  addcnul1  4453  dm0  4919  dmeq0  4923  co01  5094  clos1nrel  5887  ncprc  6125