Theorem dif1o 7580

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 7572	. . . 4 ⊢ 1𝑜 = {∅}
2	1	difeq2i 3725	. . 3 ⊢ (𝐵 ∖ 1𝑜) = (𝐵 ∖ {∅})
3	2	eleq2i 2693	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4		eldifsn 4317	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
5	3, 4	bitri 264	1 ⊢ (𝐴 ∈ (𝐵 ∖ 1𝑜) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1990   ≠ wne 2794   ∖ cdif 3571  ∅c0 3915  {csn 4177  1_𝑜c1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-suc 5729  df-1o 7560

This theorem is referenced by:  ondif1  7581  brwitnlem  7587  oelim2  7675  oeeulem  7681  oeeui  7682  omabs  7727  cantnfp1lem3  8577  cantnfp1  8578  cantnflem1  8586  cantnflem3  8588  cantnflem4  8589  cnfcom3lem  8600