Theorem el1o 8405

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 8387	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2823	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5240	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4613	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 275	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∅c0 4278  {csn 4571  1_oc1o 8373

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 5239

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 4279  df-sn 4572  df-suc 6307  df-1o 8380

This theorem is referenced by:  ord1eln01  8406  ord2eln012  8407  0lt1o  8414  oelim2  8505  oeeulem  8511  oaabs2  8559  cantnff  9559  cnfcom3lem  9588  cfsuc  10143  pf1ind  22265  mavmul0  22462  cramer0  22600  cantnfresb  43357  omabs2  43365  omcl3g  43367  f1omoOLD  48925  isinito3  49532