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Theorem n0i 3307
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2649. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i (𝐵𝐴 → ¬ 𝐴 = ∅)

Proof of Theorem n0i
StepHypRef Expression
1 noel 3306 . . 3 ¬ 𝐵 ∈ ∅
2 eleq2 2158 . . 3 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 638 . 2 (𝐴 = ∅ → ¬ 𝐵𝐴)
43con2i 595 1 (𝐵𝐴 → ¬ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1296  wcel 1445  c0 3302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015  df-nul 3303
This theorem is referenced by:  ne0i  3308  n0ii  3310  unidif0  4023  iin0r  4025  nnm00  6328  dif1enen  6676  enq0tr  7090
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