Theorem el1o 7466

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 7459	. . 3 ⊢ 1𝑜 = {∅}
2	1	eleq2i 2680	. 2 ⊢ (𝐴 ∈ 1𝑜 ↔ 𝐴 ∈ {∅})
3		0ex 4718	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4158	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 263	1 ⊢ (𝐴 ∈ 1𝑜 ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∅c0 3874  {csn 4125  1_𝑜c1o 7440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-suc 5646  df-1o 7447

This theorem is referenced by:  0lt1o  7471  oelim2  7562  oeeulem  7568  oaabs2  7612  map0e  7781  map1  7921  cantnff  8454  cnfcom3lem  8483  cfsuc  8962  pf1ind  19540  mavmul0  20177  cramer0  20315