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Theorem dif1o 6137
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 6129 . . . 4 1𝑜 = {∅}
21difeq2i 3101 . . 3 (𝐵 ∖ 1𝑜) = (𝐵 ∖ {∅})
32eleq2i 2151 . 2 (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4 eldifsn 3544 . 2 (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴𝐵𝐴 ≠ ∅))
53, 4bitri 182 1 (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴𝐵𝐴 ≠ ∅))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wcel 1436  wne 2251  cdif 2983  c0 3272  {csn 3425  1𝑜c1o 6109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rab 2364  df-v 2616  df-dif 2988  df-un 2990  df-nul 3273  df-sn 3431  df-suc 4165  df-1o 6116
This theorem is referenced by: (None)
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