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Theorem el1o 7539
 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 7532 . . 3 1𝑜 = {∅}
21eleq2i 2690 . 2 (𝐴 ∈ 1𝑜𝐴 ∈ {∅})
3 0ex 4760 . . 3 ∅ ∈ V
43elsn2 4189 . 2 (𝐴 ∈ {∅} ↔ 𝐴 = ∅)
52, 4bitri 264 1 (𝐴 ∈ 1𝑜𝐴 = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1480   ∈ wcel 1987  ∅c0 3897  {csn 4155  1𝑜c1o 7513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-dif 3563  df-un 3565  df-nul 3898  df-sn 4156  df-suc 5698  df-1o 7520 This theorem is referenced by:  0lt1o  7544  oelim2  7635  oeeulem  7641  oaabs2  7685  map0e  7855  map1  7996  cantnff  8531  cnfcom3lem  8560  cfsuc  9039  pf1ind  19659  mavmul0  20298  cramer0  20436
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