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Theorem dif1o 7525
 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 7517 . . . 4 1𝑜 = {∅}
21difeq2i 3703 . . 3 (𝐵 ∖ 1𝑜) = (𝐵 ∖ {∅})
32eleq2i 2690 . 2 (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4 eldifsn 4287 . 2 (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴𝐵𝐴 ≠ ∅))
53, 4bitri 264 1 (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴𝐵𝐴 ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1987   ≠ wne 2790   ∖ cdif 3552  ∅c0 3891  {csn 4148  1𝑜c1o 7498 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-suc 5688  df-1o 7505 This theorem is referenced by:  ondif1  7526  brwitnlem  7532  oelim2  7620  oeeulem  7626  oeeui  7627  omabs  7672  cantnfp1lem3  8521  cantnfp1  8522  cantnflem1  8530  cantnflem3  8532  cantnflem4  8533  cnfcom3lem  8544
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