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Theorem oewordi 6673
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 4481 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
2 ordgt0ge1 6580 . . . . . 6  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  1o  C_  C ) )
31, 2syl 15 . . . . 5  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  1o  C_  C
) )
4 1on 6570 . . . . . 6  |-  1o  e.  On
5 onsseleq 4512 . . . . . 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 6585 . . . . . . 7  |-  ( C  e.  ( On  \  2o )  <->  ( C  e.  On  /\  1o  e.  C ) )
10 oeword 6672 . . . . . . . . 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 6627 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
1716adantr 451 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  1o )
18 oe1m 6627 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
1918adantl 452 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  B
)  =  1o )
2017, 19eqtr4d 2393 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  ( 1o 
^o  B ) )
21 eqimss 3306 . . . . . . . 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 5949 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  A )  =  ( C  ^o  A
) )
24 oveq1 5949 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  B )  =  ( C  ^o  B
) )
2523, 24sseq12d 3283 . . . . . . 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 1642    e. wcel 1710    \ cdif 3225    C_ wss 3228   (/)c0 3531   Ord word 4470   Oncon0 4471  (class class class)co 5942   1oc1o 6556   2oc2o 6557    ^o coe 6562
This theorem is referenced by:  oelim2  6677  oeoalem  6678  oeoelem  6680  oaabs2  6727  cantnflt  7460  cnfcom  7490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-omul 6568  df-oexp 6569
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