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Theorem ne0i 3224
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2564. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
ne0i (B AA ≠ ∅)

Proof of Theorem ne0i
StepHypRef Expression
1 n0i 3223 . 2 (B A → ¬ A = ∅)
21neneqad 2278 1 (B AA ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  wne 2201  c0 3218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-nul 3219
This theorem is referenced by:  vn0  3225  inelcm  3276  rzal  3312  rexn0  3313  snnzg  3475  prnz  3480  tpnz  3483  onn0  4102  nn0eln0  4283  ordge1n0im  5951  nnmord  6019  addclpi  6304  mulclpi  6305  uzn0  8213  iccsupr  8553
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