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Theorem oewordi 7242
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 4878 . . . . . 6  |-  ( C  e.  On  ->  Ord  C )
2 ordgt0ge1 7149 . . . . . 6  |-  ( Ord 
C  ->  ( (/)  e.  C  <->  1o  C_  C ) )
31, 2syl 16 . . . . 5  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  1o  C_  C
) )
4 1on 7139 . . . . . 6  |-  1o  e.  On
5 onsseleq 4909 . . . . . 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 1020 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( (/) 
e.  C  <->  ( 1o  e.  C  \/  1o  =  C ) ) )
9 ondif2 7154 . . . . . . 7  |-  ( C  e.  ( On  \  2o )  <->  ( C  e.  On  /\  1o  e.  C ) )
10 oeword 7241 . . . . . . . . 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 1199 . . . . . . 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 1194 . . . 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 7196 . . . . . . . . . 10  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
1716adantr 465 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  1o )
18 oe1m 7196 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( 1o  ^o  B )  =  1o )
1918adantl 466 . . . . . . . . 9  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  B
)  =  1o )
2017, 19eqtr4d 2487 . . . . . . . 8  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( 1o  ^o  A
)  =  ( 1o 
^o  B ) )
21 eqimss 3541 . . . . . . . 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 6288 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  A )  =  ( C  ^o  A
) )
24 oveq1 6288 . . . . . . . 8  |-  ( 1o  =  C  ->  ( 1o  ^o  B )  =  ( C  ^o  B
) )
2523, 24sseq12d 3518 . . . . . . 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 1017 . . . . 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 974    = wceq 1383    e. wcel 1804    \ cdif 3458    C_ wss 3461   (/)c0 3770   Ord word 4867   Oncon0 4868  (class class class)co 6281   1oc1o 7125   2oc2o 7126    ^o coe 7131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138
This theorem is referenced by:  oelim2  7246  oeoalem  7247  oeoelem  7249  oaabs2  7296  cantnflt  8094  cantnfltOLD  8124  cnfcom  8147  cnfcomOLD  8155
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