Theorem el1o 8462

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 8444	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2821	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5265	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4632	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 275	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∅c0 4299  {csn 4592  1_oc1o 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-nul 4300  df-sn 4593  df-suc 6341  df-1o 8437

This theorem is referenced by:  ord1eln01  8463  ord2eln012  8464  0lt1o  8471  oelim2  8562  oeeulem  8568  oaabs2  8616  cantnff  9634  cnfcom3lem  9663  cfsuc  10217  pf1ind  22249  mavmul0  22446  cramer0  22584  cantnfresb  43320  omabs2  43328  omcl3g  43330  f1omoOLD  48886  isinito3  49493