Theorem el1o 7845

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 7838	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2897	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5013	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4431	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 267	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 198   = wceq 1658   ∈ wcel 2166  ∅c0 4143  {csn 4396  1_oc1o 7818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-nul 5012

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-v 3415  df-dif 3800  df-un 3802  df-nul 4144  df-sn 4397  df-suc 5968  df-1o 7825

This theorem is referenced by:  0lt1o  7850  oelim2  7941  oeeulem  7947  oaabs2  7991  map0eOLD  8160  cantnff  8847  cnfcom3lem  8876  cfsuc  9393  pf1ind  20078  mavmul0  20725  cramer0  20865