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Theorem oeord 6472
Description: Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oeord  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )

Proof of Theorem oeord
StepHypRef Expression
1 oeordi 6471 . . 3  |-  ( ( B  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( A  e.  B  ->  ( C  ^o  A
)  e.  ( C  ^o  B ) ) )
213adant1 978 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  ->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
3 oveq2 5718 . . . . . 6  |-  ( A  =  B  ->  ( C  ^o  A )  =  ( C  ^o  B
) )
43a1i 12 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  =  B  ->  ( C  ^o  A )  =  ( C  ^o  B ) ) )
5 oeordi 6471 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( B  e.  A  ->  ( C  ^o  B
)  e.  ( C  ^o  A ) ) )
653adant2 979 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( B  e.  A  ->  ( C  ^o  B )  e.  ( C  ^o  A ) ) )
74, 6orim12d 814 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( A  =  B  \/  B  e.  A
)  ->  ( ( C  ^o  A )  =  ( C  ^o  B
)  \/  ( C  ^o  B )  e.  ( C  ^o  A
) ) ) )
87con3d 127 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( -.  ( ( C  ^o  A )  =  ( C  ^o  B )  \/  ( C  ^o  B )  e.  ( C  ^o  A ) )  ->  -.  ( A  =  B  \/  B  e.  A )
) )
9 eldifi 3215 . . . . . 6  |-  ( C  e.  ( On  \  2o )  ->  C  e.  On )
1093ad2ant3 983 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  C  e.  On )
11 simp1 960 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  A  e.  On )
12 oecl 6422 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
1310, 11, 12syl2anc 645 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  A )  e.  On )
14 simp2 961 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  B  e.  On )
15 oecl 6422 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1610, 14, 15syl2anc 645 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  B )  e.  On )
17 eloni 4295 . . . . 5  |-  ( ( C  ^o  A )  e.  On  ->  Ord  ( C  ^o  A ) )
18 eloni 4295 . . . . 5  |-  ( ( C  ^o  B )  e.  On  ->  Ord  ( C  ^o  B ) )
19 ordtri2 4320 . . . . 5  |-  ( ( Ord  ( C  ^o  A )  /\  Ord  ( C  ^o  B ) )  ->  ( ( C  ^o  A )  e.  ( C  ^o  B
)  <->  -.  ( ( C  ^o  A )  =  ( C  ^o  B
)  \/  ( C  ^o  B )  e.  ( C  ^o  A
) ) ) )
2017, 18, 19syl2an 465 . . . 4  |-  ( ( ( C  ^o  A
)  e.  On  /\  ( C  ^o  B )  e.  On )  -> 
( ( C  ^o  A )  e.  ( C  ^o  B )  <->  -.  ( ( C  ^o  A )  =  ( C  ^o  B )  \/  ( C  ^o  B )  e.  ( C  ^o  A ) ) ) )
2113, 16, 20syl2anc 645 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  e.  ( C  ^o  B )  <->  -.  (
( C  ^o  A
)  =  ( C  ^o  B )  \/  ( C  ^o  B
)  e.  ( C  ^o  A ) ) ) )
22 eloni 4295 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
23 eloni 4295 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
24 ordtri2 4320 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2522, 23, 24syl2an 465 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A
) ) )
26253adant3 980 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
278, 21, 263imtr4d 261 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  e.  ( C  ^o  B )  ->  A  e.  B )
)
282, 27impbid 185 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ w3a 939    = wceq 1619    e. wcel 1621    \ cdif 3075   Ord word 4284   Oncon0 4285  (class class class)co 5710   2oc2o 6359    ^o coe 6364
This theorem is referenced by:  oeword  6474  oeeui  6486  omabs  6531  cantnflem3  7277
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-oexp 6371
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