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Theorem dif1o 8108
 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 8099 . . . 4 1o = {∅}
21difeq2i 4047 . . 3 (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
32eleq2i 2881 . 2 (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4 eldifsn 4680 . 2 (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴𝐵𝐴 ≠ ∅))
53, 4bitri 278 1 (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴𝐵𝐴 ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3878  ∅c0 4243  {csn 4525  1oc1o 8078 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-suc 6165  df-1o 8085 This theorem is referenced by:  ondif1  8109  brwitnlem  8115  oelim2  8204  oeeulem  8210  oeeui  8211  omabs  8257  cantnfp1lem3  9127  cantnfp1  9128  cantnflem1  9136  cantnflem3  9138  cantnflem4  9139  cnfcom3lem  9150
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