Theorem dif1o 7864

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 7856	. . . 4 ⊢ 1o = {∅}
2	1	difeq2i 3948	. . 3 ⊢ (𝐵 ∖ 1o) = (𝐵 ∖ {∅})
3	2	eleq2i 2851	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ 𝐴 ∈ (𝐵 ∖ {∅}))
4		eldifsn 4550	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
5	3, 4	bitri 267	1 ⊢ (𝐴 ∈ (𝐵 ∖ 1o) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))

Syntax hints:   ↔ wb 198   ∧ wa 386   ∈ wcel 2107   ≠ wne 2969   ∖ cdif 3789  ∅c0 4141  {csn 4398  1_oc1o 7836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-nul 4142  df-sn 4399  df-suc 5982  df-1o 7843

This theorem is referenced by:  ondif1  7865  brwitnlem  7871  oelim2  7959  oeeulem  7965  oeeui  7966  omabs  8011  cantnfp1lem3  8874  cantnfp1  8875  cantnflem1  8883  cantnflem3  8885  cantnflem4  8886  cnfcom3lem  8897