MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe0m1 Unicode version

Theorem oe0m1 6756
Description: Ordinal exponentiation with zero mantissa and nonzero exponent. Proposition 8.31(2) of [TakeutiZaring] p. 67 and its converse. (Contributed by NM, 5-Jan-2005.)
Assertion
Ref Expression
oe0m1  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  ( (/)  ^o  A
)  =  (/) ) )

Proof of Theorem oe0m1
StepHypRef Expression
1 eloni 4583 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordgt0ge1 6732 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
31, 2syl 16 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  1o  C_  A
) )
4 oe0m 6753 . . . 4  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )
54eqeq1d 2443 . . 3  |-  ( A  e.  On  ->  (
( (/)  ^o  A )  =  (/)  <->  ( 1o  \  A )  =  (/) ) )
6 ssdif0 3678 . . 3  |-  ( 1o  C_  A  <->  ( 1o  \  A )  =  (/) )
75, 6syl6rbbr 256 . 2  |-  ( A  e.  On  ->  ( 1o  C_  A  <->  ( (/)  ^o  A
)  =  (/) ) )
83, 7bitrd 245 1  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  ( (/)  ^o  A
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    \ cdif 3309    C_ wss 3312   (/)c0 3620   Ord word 4572   Oncon0 4573  (class class class)co 6072   1oc1o 6708    ^o coe 6714
This theorem is referenced by:  oev2  6758  oesuclem  6760  oecl  6772  oewordri  6826  oelim2  6829  oeoa  6831  oeoe  6833  cantnf  7638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-iota 5409  df-fun 5447  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-recs 6624  df-rdg 6659  df-1o 6715  df-oexp 6721
  Copyright terms: Public domain W3C validator