Theorem el1o 8509

Description: Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)

Assertion

Ref	Expression
el1o	⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Proof of Theorem el1o

Step	Hyp	Ref	Expression
1		df1o2 8487	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2820	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5301	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4663	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 275	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 205   = wceq 1534   ∈ wcel 2099  ∅c0 4318  {csn 4624  1_oc1o 8473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-dif 3947  df-un 3949  df-nul 4319  df-sn 4625  df-suc 6369  df-1o 8480

This theorem is referenced by:  ord1eln01  8510  ord2eln012  8511  0lt1o  8518  oelim2  8609  oeeulem  8615  oaabs2  8663  cantnff  9689  cnfcom3lem  9718  cfsuc  10272  pf1ind  22261  mavmul0  22441  cramer0  22579  cantnfresb  42676  omabs2  42684  omcl3g  42686  f1omo  47836