MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  el1o Structured version   Visualization version   GIF version

Theorem el1o 7737
Description: Membership in ordinal one. (Contributed by NM, 5-Jan-2005.)
Assertion
Ref Expression
el1o (𝐴 ∈ 1𝑜𝐴 = ∅)

Proof of Theorem el1o
StepHypRef Expression
1 df1o2 7730 . . 3 1𝑜 = {∅}
21eleq2i 2842 . 2 (𝐴 ∈ 1𝑜𝐴 ∈ {∅})
3 0ex 4925 . . 3 ∅ ∈ V
43elsn2 4351 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 264 1 (𝐴 ∈ 1𝑜𝐴 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wcel 2145  c0 4063  {csn 4317  1𝑜c1o 7710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4318  df-suc 5871  df-1o 7717
This theorem is referenced by:  0lt1o  7742  oelim2  7833  oeeulem  7839  oaabs2  7883  map0eOLD  8052  cantnff  8739  cnfcom3lem  8768  cfsuc  9285  pf1ind  19934  mavmul0  20576  cramer0  20716
  Copyright terms: Public domain W3C validator