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Theorem ordgt0ge1 6382
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 4352 . . 3  |-  (/)  e.  On
2 ordelsuc 4464 . . 3  |-  ( (
(/)  e.  On  /\  Ord  A )  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
31, 2mpan 421 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
4 df-1o 6363 . . 3  |-  1o  =  suc  (/)
54sseq1i 3154 . 2  |-  ( 1o  C_  A  <->  suc  (/)  C_  A )
63, 5bitr4di 197 1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 2128    C_ wss 3102   (/)c0 3394   Ord word 4322   Oncon0 4323   suc csuc 4325   1oc1o 6356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-1o 6363
This theorem is referenced by:  ordge1n0im  6383  archnqq  7337
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