Theorem el1o 8541

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 8521	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5267	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4627	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 278	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∅c0 4281  {csn 4588  1_oc1o 8507

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5266

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-suc 6394  df-1o 8514

This theorem is used by:  ord1eln01  8542  ord2eln012  8543  0lt1o  8550  oelim2  8642  oeeulem  8648  oaabs2  8696  cantnff  9712  cnfcom3lem  9741  cfsuc  10374  pf1ind  22728  mavmul0  22922  cramer0  23063  selvply1rhmlem2  34264  cantnfresb  44587  omabs2  44595  omcl3g  44597  f1omoOLD  50339  isinito3  50966