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Theorem ordgt0ge1 6181
Description: Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
ordgt0ge1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )

Proof of Theorem ordgt0ge1
StepHypRef Expression
1 0elon 4210 . . 3  |-  (/)  e.  On
2 ordelsuc 4312 . . 3  |-  ( (
(/)  e.  On  /\  Ord  A )  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
31, 2mpan 415 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
4 df-1o 6163 . . 3  |-  1o  =  suc  (/)
54sseq1i 3048 . 2  |-  ( 1o  C_  A  <->  suc  (/)  C_  A )
63, 5syl6bbr 196 1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    e. wcel 1438    C_ wss 2997   (/)c0 3284   Ord word 4180   Oncon0 4181   suc csuc 4183   1oc1o 6156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3957
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189  df-1o 6163
This theorem is referenced by:  ordge1n0im  6182  archnqq  6955
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