ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ne0i GIF version

Theorem ne0i 3258
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2587. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
ne0i (𝐵𝐴𝐴 ≠ ∅)

Proof of Theorem ne0i
StepHypRef Expression
1 n0i 3257 . 2 (𝐵𝐴 → ¬ 𝐴 = ∅)
21neneqad 2299 1 (𝐵𝐴𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  wne 2220  c0 3252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2948  df-nul 3253
This theorem is referenced by:  vn0  3259  inelcm  3310  rzal  3346  rexn0  3347  snnzg  3513  prnz  3518  tpnz  3521  onn0  4165  nn0eln0  4369  ordge1n0im  6050  nnmord  6121  phpm  6358  addclpi  6483  mulclpi  6484  uzn0  8584  iccsupr  8936
  Copyright terms: Public domain W3C validator