Theorem el1o 8532

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 8512	. . 3 ⊢ 1o = {∅}
2	1	eleq2i 2831	. 2 ⊢ (𝐴 ∈ 1o ↔ 𝐴 ∈ {∅})
3		0ex 5313	. . 3 ⊢ ∅ ∈ V
4	3	elsn2 4670	. 2 ⊢ (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
5	2, 4	bitri 275	1 ⊢ (𝐴 ∈ 1o ↔ 𝐴 = ∅)

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ∅c0 4339  {csn 4631  1_oc1o 8498

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-suc 6392  df-1o 8505

This theorem is referenced by:  ord1eln01  8533  ord2eln012  8534  0lt1o  8541  oelim2  8632  oeeulem  8638  oaabs2  8686  cantnff  9712  cnfcom3lem  9741  cfsuc  10295  pf1ind  22375  mavmul0  22574  cramer0  22712  cantnfresb  43314  omabs2  43322  omcl3g  43324  f1omo  48691