Theorem dif1o 8128

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 8119	. . . 4 ⊢ 1o = {∅}
2	1	difeq2i 4099	. . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
3	2	eleq2i 2907	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4		eldifsn 4722	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
5	3, 4	bitri 277	1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2113   ≠ wne 3019   ∖ cdif 3936  ∅c0 4294  {csn 4570  1_oc1o 8098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4571  df-suc 6200  df-1o 8105

This theorem is referenced by:  ondif1  8129  brwitnlem  8135  oelim2  8224  oeeulem  8230  oeeui  8231  omabs  8277  cantnfp1lem3  9146  cantnfp1  9147  cantnflem1  9155  cantnflem3  9157  cantnflem4  9158  cnfcom3lem  9169