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Theorem oewordi 7130
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 4827 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
2 ordgt0ge1 7037 . . . . . 6  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  1o  C_  C ) )
31, 2syl 16 . . . . 5  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  1o  C_  C
) )
4 1on 7027 . . . . . 6  |-  1o  e.  On
5 onsseleq 4858 . . . . . 6  |-  ( ( 1o  e.  On  /\  C  e.  On )  ->  ( 1o  C_  C  <->  ( 1o  e.  C  \/  1o  =  C )
) )
64, 5mpan 670 . . . . 5  |-  ( C  e.  On  ->  ( 1o  C_  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
73, 6bitrd 253 . . . 4  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
873ad2ant3 1011 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
9 ondif2 7042 . . . . . . 7  |-  ( C  e.  ( On  \  2o )  <->  ( C  e.  On  /\  1o  e.  C ) )
10 oeword 7129 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
1110biimpd 207 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
12113expia 1190 . . . . . . 7  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  e.  ( On  \  2o )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
139, 12syl5bir 218 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( C  e.  On  /\  1o  e.  C )  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
1413expd 436 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( C  e.  On  ->  ( 1o  e.  C  ->  ( A  C_  B  ->  ( C  ^o  A
)  C_  ( C  ^o  B ) ) ) ) )
15143impia 1185 . . . 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 7084 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
1716adantr 465 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  1o )
18 oe1m 7084 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
1918adantl 466 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  B
)  =  1o )
2017, 19eqtr4d 2495 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  ( 1o 
^o  B ) )
21 eqimss 3506 . . . . . . . 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 6197 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  A )  =  ( C  ^o  A
) )
24 oveq1 6197 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  B )  =  ( C  ^o  B
) )
2523, 24sseq12d 3483 . . . . . . 7  |-  ( 1o  =  C  ->  (
( 1o  ^o  A
)  C_  ( 1o  ^o  B )  <->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
2622, 25syl5ibcom 220 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  =  C  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) )
27263adant3 1008 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( 1o  =  C  ->  ( C  ^o  A ) 
C_  ( C  ^o  B ) ) )
2827a1dd 46 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( 1o  =  C  ->  ( A  C_  B  ->  ( C  ^o  A ) 
C_  ( C  ^o  B ) ) ) )
2915, 28jaod 380 . . 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 215 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  ->  ( A  C_  B  ->  ( C  ^o  A )  C_  ( C  ^o  B ) ) ) )
3130imp 429 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    \ cdif 3423    C_ wss 3426   (/)c0 3735   Ord word 4816   Oncon0 4817  (class class class)co 6190   1oc1o 7013   2oc2o 7014    ^o coe 7019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-omul 7025  df-oexp 7026
This theorem is referenced by:  oelim2  7134  oeoalem  7135  oeoelem  7137  oaabs2  7184  cantnflt  7981  cantnfltOLD  8011  cnfcom  8034  cnfcomOLD  8042
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