Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dif1o Structured version   Visualization version   GIF version

Theorem dif1o 7734
 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 7726 . . . 4 1𝑜 = {∅}
21difeq2i 3876 . . 3 (𝐵 ∖ 1𝑜) = (𝐵 ∖ {∅})
32eleq2i 2842 . 2 (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4 eldifsn 4453 . 2 (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴𝐵𝐴 ≠ ∅))
53, 4bitri 264 1 (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴𝐵𝐴 ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 382   ∈ wcel 2145   ≠ wne 2943   ∖ cdif 3720  ∅c0 4063  {csn 4316  1𝑜c1o 7706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4317  df-suc 5872  df-1o 7713 This theorem is referenced by:  ondif1  7735  brwitnlem  7741  oelim2  7829  oeeulem  7835  oeeui  7836  omabs  7881  cantnfp1lem3  8741  cantnfp1  8742  cantnflem1  8750  cantnflem3  8752  cantnflem4  8753  cnfcom3lem  8764
 Copyright terms: Public domain W3C validator