MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ne0ii Structured version   Visualization version   GIF version

Theorem ne0ii 4071
Description: If a set has elements, then it is not empty. Inference associated with ne0i 4069. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
n0ii.1 𝐴𝐵
Assertion
Ref Expression
ne0ii 𝐵 ≠ ∅

Proof of Theorem ne0ii
StepHypRef Expression
1 n0ii.1 . 2 𝐴𝐵
2 ne0i 4069 . 2 (𝐴𝐵𝐵 ≠ ∅)
31, 2ax-mp 5 1 𝐵 ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  wne 2943  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by:  vn0  4072  prnz  4445  tpnz  4447  pwne0  4966  onn0  5932  oawordeulem  7788  noinfep  8721  fin23lem31  9367  isfin1-3  9410  omina  9715  rpnnen1lem4  12020  rpnnen1lem5  12021  rpnnen1lem4OLD  12026  rpnnen1lem5OLD  12027  rexfiuz  14295  caurcvg  14615  caurcvg2  14616  caucvg  14617  infcvgaux1i  14796  divalglem2  15326  pc2dvds  15790  vdwmc2  15890  cnsubglem  20010  cnmsubglem  20024  pmatcollpw3  20809  zfbas  21920  nrginvrcn  22716  lebnumlem3  22982  caun0  23298  cnflduss  23371  cnfldcusp  23372  reust  23388  recusp  23389  nulmbl2  23524  itg2seq  23729  itg2monolem1  23737  c1lip1  23980  aannenlem2  24304  logbmpt  24747  tgcgr4  25647  shintcl  28529  chintcl  28531  nmoprepnf  29066  nmfnrepnf  29079  nmcexi  29225  snct  29831  esum0  30451  esumpcvgval  30480  bnj906  31338  bj-tagn0  33298  taupi  33506  ismblfin  33783  volsupnfl  33787  itg2addnclem  33793  ftc1anc  33825  incsequz  33876  isbnd3  33915  ssbnd  33919  corclrcl  38525  imo72b2lem0  38991  imo72b2lem2  38993  imo72b2lem1  38997  imo72b2  39001  amgm2d  39027  nnn0  40111  ren0  40142  ioodvbdlimc1  40666  ioodvbdlimc2  40668  stirlinglem13  40820  fourierdlem103  40943  fourierdlem104  40944  fouriersw  40965  hoicvr  41282  2zlidl  42462
  Copyright terms: Public domain W3C validator