Theorem el1o 8118

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 8110	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2904	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5203	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4597	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 277	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∅c0 4290  {csn 4560  1_oc1o 8089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4561  df-suc 6191  df-1o 8096

This theorem is referenced by:  0lt1o  8123  oelim2  8215  oeeulem  8221  oaabs2  8266  cantnff  9131  cnfcom3lem  9160  cfsuc  9673  pf1ind  20512  mavmul0  21155  cramer0  21293