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Theorem n0i 3419
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2746. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i (𝐵𝐴 → ¬ 𝐴 = ∅)

Proof of Theorem n0i
StepHypRef Expression
1 noel 3418 . . 3 ¬ 𝐵 ∈ ∅
2 eleq2 2234 . . 3 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 670 . 2 (𝐴 = ∅ → ¬ 𝐵𝐴)
43con2i 622 1 (𝐵𝐴 → ¬ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1348  wcel 2141  c0 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415
This theorem is referenced by:  ne0i  3420  n0ii  3422  unidif0  4151  iin0r  4153  nnm00  6506  dif1enen  6855  enq0tr  7385
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