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Theorem oe0m1 7163
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 4877 . . 3  |-  ( A  e.  On  ->  Ord  A )
2 ordgt0ge1 7139 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  1o  C_  A ) )
31, 2syl 16 . 2  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  1o  C_  A
) )
4 oe0m 7160 . . . 4  |-  ( A  e.  On  ->  ( (/) 
^o  A )  =  ( 1o  \  A
) )
54eqeq1d 2456 . . 3  |-  ( A  e.  On  ->  (
( (/)  ^o  A )  =  (/)  <->  ( 1o  \  A )  =  (/) ) )
6 ssdif0 3873 . . 3  |-  ( 1o  C_  A  <->  ( 1o  \  A )  =  (/) )
75, 6syl6rbbr 264 . 2  |-  ( A  e.  On  ->  ( 1o  C_  A  <->  ( (/)  ^o  A
)  =  (/) ) )
83, 7bitrd 253 1  |-  ( A  e.  On  ->  ( (/) 
e.  A  <->  ( (/)  ^o  A
)  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   (/)c0 3783   Ord word 4866   Oncon0 4867  (class class class)co 6270   1oc1o 7115    ^o coe 7121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-1o 7122  df-oexp 7128
This theorem is referenced by:  oev2  7165  oesuclem  7167  oecl  7179  oewordri  7233  oelim2  7236  oeoa  7238  oeoe  7240  cantnf  8103  cantnfOLD  8125
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