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

Theorem ordgt0ge1 7165
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 4940 . . 3  |-  (/)  e.  On
2 ordelsuc 6654 . . 3  |-  ( (
(/)  e.  On  /\  Ord  A )  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
31, 2mpan 670 . 2  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  suc  (/)  C_  A ) )
4 df-1o 7148 . . 3  |-  1o  =  suc  (/)
54sseq1i 3523 . 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 1819    C_ wss 3471   (/)c0 3793   Ord word 4886   Oncon0 4887   suc csuc 4889   1oc1o 7141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-1o 7148
This theorem is referenced by:  ordge1n0  7166  oe0m1  7189  omword1  7240  omword2  7241  omlimcl  7245  oen0  7253  oewordi  7258
  Copyright terms: Public domain W3C validator