Theorem el1o 8107

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 8099	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2881	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5175	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4564	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 278	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∅c0 4243  {csn 4525  1_oc1o 8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-suc 6165  df-1o 8085

This theorem is referenced by:  0lt1o  8112  oelim2  8204  oeeulem  8210  oaabs2  8255  cantnff  9121  cnfcom3lem  9150  cfsuc  9668  pf1ind  20979  mavmul0  21157  cramer0  21295