Theorem n0i 3556

Description: If a set has elements, it is not empty. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
n0i	⊢ (B ∈ A → ¬ A = ∅)

Proof of Theorem n0i

Step	Hyp	Ref	Expression
1		noel 3555	. . 3 ⊢ ¬ B ∈ ∅
2		eleq2 2414	. . 3 ⊢ (A = ∅ → (B ∈ A ↔ B ∈ ∅))
3	1, 2	mtbiri 294	. 2 ⊢ (A = ∅ → ¬ B ∈ A)
4	3	con2i 112	1 ⊢ (B ∈ A → ¬ A = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = 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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  ne0i  3557  0nelsuc  4401  nndisjeq  4430  nnceleq  4431  sfinltfin  4536  vfin1cltv  4548  funiunfv  5468  ecexr  5951  nceleq  6150  1p1e2c  6156  2p1e3c  6157