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Theorem oewordi 6793
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 4551 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
2 ordgt0ge1 6700 . . . . . 6  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  1o  C_  C ) )
31, 2syl 16 . . . . 5  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  1o  C_  C
) )
4 1on 6690 . . . . . 6  |-  1o  e.  On
5 onsseleq 4582 . . . . . 6  |-  ( ( 1o  e.  On  /\  C  e.  On )  ->  ( 1o  C_  C  <->  ( 1o  e.  C  \/  1o  =  C )
) )
64, 5mpan 652 . . . . 5  |-  ( C  e.  On  ->  ( 1o  C_  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
73, 6bitrd 245 . . . 4  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
873ad2ant3 980 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
9 ondif2 6705 . . . . . . 7  |-  ( C  e.  ( On  \  2o )  <->  ( C  e.  On  /\  1o  e.  C ) )
10 oeword 6792 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
1110biimpd 199 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
12113expia 1155 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  e.  ( On  \  2o )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
139, 12syl5bir 210 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( C  e.  On  /\  1o  e.  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
1413exp3a 426 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  e.  On  ->  ( 1o  e.  C  ->  ( A  C_  B  ->  ( C  ^o  A
)  C_  ( C  ^o  B ) ) ) ) )
15143impia 1150 . . . 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 6747 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
1716adantr 452 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  1o )
18 oe1m 6747 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
1918adantl 453 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  B
)  =  1o )
2017, 19eqtr4d 2439 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  ( 1o 
^o  B ) )
21 eqimss 3360 . . . . . . . 8  |-  ( ( 1o  ^o  A )  =  ( 1o  ^o  B )  ->  ( 1o  ^o  A )  C_  ( 1o  ^o  B ) )
2220, 21syl 16 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  C_  ( 1o  ^o  B ) )
23 oveq1 6047 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  A )  =  ( C  ^o  A
) )
24 oveq1 6047 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  B )  =  ( C  ^o  B
) )
2523, 24sseq12d 3337 . . . . . . 7  |-  ( 1o  =  C  ->  (
( 1o  ^o  A
)  C_  ( 1o  ^o  B )  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
2622, 25syl5ibcom 212 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  =  C  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
27263adant3 977 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( 1o  =  C  ->  ( C  ^o  A ) 
C_  ( C  ^o  B ) ) )
2827a1dd 44 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( 1o  =  C  ->  ( A  C_  B  ->  ( C  ^o  A ) 
C_  ( C  ^o  B ) ) ) )
2915, 28jaod 370 . . 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 207 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
3130imp 419 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3277    C_ wss 3280   (/)c0 3588   Ord word 4540   Oncon0 4541  (class class class)co 6040   1oc1o 6676   2oc2o 6677    ^o coe 6682
This theorem is referenced by:  oelim2  6797  oeoalem  6798  oeoelem  6800  oaabs2  6847  cantnflt  7583  cnfcom  7613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-oexp 6689
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