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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
StepHypRef Expression
1 noel 3554 . . 3 ¬ B
2 eleq2 2414 . . 3 (A = → (B AB ))
31, 2mtbiri 294 . 2 (A = → ¬ B A)
43con2i 112 1 (B A → ¬ A = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = 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-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
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