Theorem n0i 3555

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 3554	. . 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 3550

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by:  ne0i  3556  0nelsuc  4400  nndisjeq  4429  nnceleq  4430  sfinltfin  4535  vfin1cltv  4547  funiunfv  5467  ecexr  5950  nceleq  6149  1p1e2c  6155  2p1e3c  6156