Theorem dif1o 8556

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 8529	. . . 4 ⊢ 1o = {∅}
2	1	difeq2i 4146	. . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
3	2	eleq2i 2836	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4		eldifsn 4811	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
5	3, 4	bitri 275	1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973  ∅c0 4352  {csn 4648  1_oc1o 8515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-suc 6401  df-1o 8522

This theorem is referenced by:  ondif1  8557  brwitnlem  8563  oelim2  8651  oeeulem  8657  oeeui  8658  omabs  8707  cantnfp1lem3  9749  cantnfp1  9750  cantnflem1  9758  cantnflem3  9760  cantnflem4  9761  cnfcom3lem  9772