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Theorem dif1o 8450
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 8423 . . . 4 1o = {∅}
21difeq2i 4083 . . 3 (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
32eleq2i 2826 . 2 (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4 eldifsn 4751 . 2 (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴𝐵𝐴 ≠ ∅))
53, 4bitri 275 1 (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴𝐵𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  wne 2940  cdif 3911  c0 4286  {csn 4590  1oc1o 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-nul 4287  df-sn 4591  df-suc 6327  df-1o 8416
This theorem is referenced by:  ondif1  8451  brwitnlem  8457  oelim2  8546  oeeulem  8552  oeeui  8553  omabs  8601  cantnfp1lem3  9624  cantnfp1  9625  cantnflem1  9633  cantnflem3  9635  cantnflem4  9636  cnfcom3lem  9647
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