Theorem el1o 8410

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 8392	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2823	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5243	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4615	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 275	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∅c0 4280  {csn 4573  1_oc1o 8378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-un 3902  df-nul 4281  df-sn 4574  df-suc 6312  df-1o 8385

This theorem is referenced by:  ord1eln01  8411  ord2eln012  8412  0lt1o  8419  oelim2  8510  oeeulem  8516  oaabs2  8564  cantnff  9564  cnfcom3lem  9593  cfsuc  10148  pf1ind  22270  mavmul0  22467  cramer0  22605  cantnfresb  43416  omabs2  43424  omcl3g  43426  f1omoOLD  48993  isinito3  49600