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Theorem ordgt0ge1 6949
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 4784 . . 3  |-  (/)  e.  On
2 ordelsuc 6443 . . 3  |-  ( (
(/)  e.  On  /\  Ord  A )  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
31, 2mpan 670 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
4 df-1o 6932 . . 3  |-  1o  =  suc  (/)
54sseq1i 3392 . 2  |-  ( 1o  C_  A  <->  suc  (/)  C_  A )
63, 5syl6bbr 263 1  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1756    C_ wss 3340   (/)c0 3649   Ord word 4730   Oncon0 4731   suc csuc 4733   1oc1o 6925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-tr 4398  df-eprel 4644  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-suc 4737  df-1o 6932
This theorem is referenced by:  ordge1n0  6950  oe0m1  6973  omword1  7024  omword2  7025  omlimcl  7029  oen0  7037  oewordi  7042
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