Theorem neq0 3561

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
neq0	⊢ (¬ A = ∅ ↔ ∃x x ∈ A)

Distinct variable group:   x,A

Proof of Theorem neq0

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 ⊢ (A ≠ ∅ ↔ ¬ A = ∅)
2		n0 3560	. 2 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
3	1, 2	bitr3i 242	1 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  ∅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:  eq0  3565  ralidm  3654  snprc  3789  pwpw0  3856  sssn  3865  pwsnALT  3883  uni0b  3917  nndisjeq  4430  isomin  5497  erdisj  5973