Theorem dif1o 7467

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) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))

Proof of Theorem dif1o

Step	Hyp	Ref	Expression
1		df1o2 7459	. . . 4 ⊢ 1𝑜 = {∅}
2	1	difeq2i 3687	. . 3 ⊢ (𝐵 ∖ 1𝑜) = (𝐵 ∖ {∅})
3	2	eleq2i 2680	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4		eldifsn 4260	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
5	3, 4	bitri 263	1 ⊢ (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537  ∅c0 3874  {csn 4125  1_𝑜c1o 7440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-suc 5646  df-1o 7447

This theorem is referenced by:  ondif1  7468  brwitnlem  7474  oelim2  7562  oeeulem  7568  oeeui  7569  omabs  7614  cantnfp1lem3  8460  cantnfp1  8461  cantnflem1  8469  cantnflem3  8471  cantnflem4  8472  cnfcom3lem  8483