Theorem dif1o 8410

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 8387	. . . 4 ⊢ 1o = {∅}
2	1	difeq2i 4068	. . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
3	2	eleq2i 2823	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4		eldifsn 4733	. 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 2111   ≠ wne 2928   ∖ cdif 3894  ∅c0 4278  {csn 4571  1_oc1o 8373

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-nul 4279  df-sn 4572  df-suc 6307  df-1o 8380

This theorem is referenced by:  ondif1  8411  brwitnlem  8417  oelim2  8505  oeeulem  8511  oeeui  8512  omabs  8561  cantnfp1lem3  9565  cantnfp1  9566  cantnflem1  9574  cantnflem3  9576  cantnflem4  9577  cnfcom3lem  9588