MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordgt0ge1 Structured version   Unicode version

Theorem ordgt0ge1 7159
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 4937 . . 3  |-  (/)  e.  On
2 ordelsuc 6650 . . 3  |-  ( (
(/)  e.  On  /\  Ord  A )  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
31, 2mpan 670 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
4 df-1o 7142 . . 3  |-  1o  =  suc  (/)
54sseq1i 3533 . 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 1767    C_ wss 3481   (/)c0 3790   Ord word 4883   Oncon0 4884   suc csuc 4886   1oc1o 7135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-1o 7142
This theorem is referenced by:  ordge1n0  7160  oe0m1  7183  omword1  7234  omword2  7235  omlimcl  7239  oen0  7247  oewordi  7252
  Copyright terms: Public domain W3C validator