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Theorem oe0m1 6516
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 4402 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordgt0ge1 6492 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
31, 2syl 17 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  1o  C_  A
) )
4 oe0m 6513 . . . 4  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )
54eqeq1d 2293 . . 3  |-  ( A  e.  On  ->  (
( (/)  ^o  A )  =  (/)  <->  ( 1o  \  A )  =  (/) ) )
6 ssdif0 3515 . . 3  |-  ( 1o  C_  A  <->  ( 1o  \  A )  =  (/) )
75, 6syl6rbbr 257 . 2  |-  ( A  e.  On  ->  ( 1o  C_  A  <->  ( (/)  ^o  A
)  =  (/) ) )
83, 7bitrd 246 1  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  ( (/)  ^o  A
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624    e. wcel 1685    \ cdif 3151    C_ wss 3154   (/)c0 3457   Ord word 4391   Oncon0 4392  (class class class)co 5820   1oc1o 6468    ^o coe 6474
This theorem is referenced by:  oev2  6518  oesuclem  6520  oecl  6532  oewordri  6586  oelim2  6589  oeoa  6591  oeoe  6593  cantnf  7391
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-1o 6475  df-oexp 6481
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