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Theorem oe0m1 6474
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 4360 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordgt0ge1 6450 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
31, 2syl 17 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  1o  C_  A
) )
4 oe0m 6471 . . . 4  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )
54eqeq1d 2264 . . 3  |-  ( A  e.  On  ->  (
( (/)  ^o  A )  =  (/)  <->  ( 1o  \  A )  =  (/) ) )
6 ssdif0 3474 . . 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 1619    e. wcel 1621    \ cdif 3110    C_ wss 3113   (/)c0 3416   Ord word 4349   Oncon0 4350  (class class class)co 5778   1oc1o 6426    ^o coe 6432
This theorem is referenced by:  oev2  6476  oesuclem  6478  oecl  6490  oewordri  6544  oelim2  6547  oeoa  6549  oeoe  6551  cantnf  7349
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-suc 4356  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-recs 6342  df-rdg 6377  df-1o 6433  df-oexp 6439
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