Theorem dif1o 8427

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 8404	. . . 4 ⊢ 1o = {∅}
2	1	difeq2i 4075	. . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
3	2	eleq2i 2828	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4		eldifsn 4742	. 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 2113   ≠ wne 2932   ∖ cdif 3898  ∅c0 4285  {csn 4580  1_oc1o 8390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-nul 4286  df-sn 4581  df-suc 6323  df-1o 8397

This theorem is referenced by:  ondif1  8428  brwitnlem  8434  oelim2  8523  oeeulem  8529  oeeui  8530  omabs  8579  cantnfp1lem3  9589  cantnfp1  9590  cantnflem1  9598  cantnflem3  9600  cantnflem4  9601  cnfcom3lem  9612