Theorem eq0 3564

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 3560	. . 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 3550

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by:  0el  3566  disj  3591  ssdif0  3609  difin0ss  3616  inssdif0  3617  ralf0  3656  addcnul1  4452  dm0  4918  dmeq0  4922  co01  5093  clos1nrel  5886  ncprc  6124