Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0ii Structured version   Visualization version   GIF version

Theorem n0ii 4053
 Description: If a set has elements, then it is not empty. Inference associated with n0i 4051. (Contributed by BJ, 15-Jul-2021.)
Hypothesis
Ref Expression
n0ii.1 𝐴𝐵
Assertion
Ref Expression
n0ii ¬ 𝐵 = ∅

Proof of Theorem n0ii
StepHypRef Expression
1 n0ii.1 . 2 𝐴𝐵
2 n0i 4051 . 2 (𝐴𝐵 → ¬ 𝐵 = ∅)
31, 2ax-mp 5 1 ¬ 𝐵 = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1620   ∈ wcel 2127  ∅c0 4046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-dif 3706  df-nul 4047 This theorem is referenced by:  iin0  4976  snsn0non  5995  tfrlem16  7646  pwcdadom  9201  nnunb  11451  hon0  28932  dmadjrnb  29045  bnj98  31215  dvnprodlem3  40635
 Copyright terms: Public domain W3C validator