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Theorem dif1o 8537
Description: Two ways to say that 𝐴 is a nonzero number of the set 𝐵. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
dif1o (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴𝐵𝐴 ≠ ∅))

Proof of Theorem dif1o
StepHypRef Expression
1 df1o2 8512 . . . 4 1o = {∅}
21difeq2i 4133 . . 3 (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
32eleq2i 2831 . 2 (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4 eldifsn 4791 . 2 (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴𝐵𝐴 ≠ ∅))
53, 4bitri 275 1 (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴𝐵𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  wne 2938  cdif 3960  c0 4339  {csn 4631  1oc1o 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-suc 6392  df-1o 8505
This theorem is referenced by:  ondif1  8538  brwitnlem  8544  oelim2  8632  oeeulem  8638  oeeui  8639  omabs  8688  cantnfp1lem3  9718  cantnfp1  9719  cantnflem1  9727  cantnflem3  9729  cantnflem4  9730  cnfcom3lem  9741
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