Theorem el1o 8476

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 8456	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2861	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5269	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4633	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 278	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∅c0 4294  {csn 4591  1_oc1o 8442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5268

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-nul 4295  df-sn 4592  df-suc 6364  df-1o 8449

This theorem is referenced by:  ord1eln01  8477  ord2eln012  8478  0lt1o  8485  oelim2  8577  oeeulem  8583  oaabs2  8631  cantnff  9639  cnfcom3lem  9668  cfsuc  10237  pf1ind  22480  mavmul0  22674  cramer0  22812  selvply1rhmlem2  33852  cantnfresb  43938  omabs2  43946  omcl3g  43948  f1omoOLD  49552  isinito3  50158