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Theorem oewordi 6589
Description: Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.)
Assertion
Ref Expression
oewordi  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )

Proof of Theorem oewordi
StepHypRef Expression
1 eloni 4402 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
2 ordgt0ge1 6496 . . . . . 6  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  1o  C_  C ) )
31, 2syl 15 . . . . 5  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  1o  C_  C
) )
4 1on 6486 . . . . . 6  |-  1o  e.  On
5 onsseleq 4433 . . . . . 6  |-  ( ( 1o  e.  On  /\  C  e.  On )  ->  ( 1o  C_  C  <->  ( 1o  e.  C  \/  1o  =  C )
) )
64, 5mpan 651 . . . . 5  |-  ( C  e.  On  ->  ( 1o  C_  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
73, 6bitrd 244 . . . 4  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
873ad2ant3 978 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
9 ondif2 6501 . . . . . . 7  |-  ( C  e.  ( On  \  2o )  <->  ( C  e.  On  /\  1o  e.  C ) )
10 oeword 6588 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
1110biimpd 198 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
12113expia 1153 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  e.  ( On  \  2o )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
139, 12syl5bir 209 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( C  e.  On  /\  1o  e.  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
1413exp3a 425 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  e.  On  ->  ( 1o  e.  C  ->  ( A  C_  B  ->  ( C  ^o  A
)  C_  ( C  ^o  B ) ) ) ) )
15143impia 1148 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( 1o  e.  C  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
16 oe1m 6543 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
1716adantr 451 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  1o )
18 oe1m 6543 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
1918adantl 452 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  B
)  =  1o )
2017, 19eqtr4d 2318 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  ( 1o 
^o  B ) )
21 eqimss 3230 . . . . . . . 8  |-  ( ( 1o  ^o  A )  =  ( 1o  ^o  B )  ->  ( 1o  ^o  A )  C_  ( 1o  ^o  B ) )
2220, 21syl 15 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  C_  ( 1o  ^o  B ) )
23 oveq1 5865 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  A )  =  ( C  ^o  A
) )
24 oveq1 5865 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  B )  =  ( C  ^o  B
) )
2523, 24sseq12d 3207 . . . . . . 7  |-  ( 1o  =  C  ->  (
( 1o  ^o  A
)  C_  ( 1o  ^o  B )  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
2622, 25syl5ibcom 211 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  =  C  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
27263adant3 975 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( 1o  =  C  ->  ( C  ^o  A ) 
C_  ( C  ^o  B ) ) )
2827a1dd 42 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( 1o  =  C  ->  ( A  C_  B  ->  ( C  ^o  A ) 
C_  ( C  ^o  B ) ) ) )
2915, 28jaod 369 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( 1o  e.  C  \/  1o  =  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
308, 29sylbid 206 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
3130imp 418 1  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  (/)  e.  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   (/)c0 3455   Ord word 4391   Oncon0 4392  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ^o coe 6478
This theorem is referenced by:  oelim2  6593  oeoalem  6594  oeoelem  6596  oaabs2  6643  cantnflt  7373  cnfcom  7403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-lim 4397  df-suc 4398  df-om 4657  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485
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