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Theorem oeord 6822
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 6821 . . 3  |-  ( ( B  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( A  e.  B  ->  ( C  ^o  A
)  e.  ( C  ^o  B ) ) )
213adant1 975 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  ->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
3 oveq2 6080 . . . . . 6  |-  ( A  =  B  ->  ( C  ^o  A )  =  ( C  ^o  B
) )
43a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  =  B  ->  ( C  ^o  A )  =  ( C  ^o  B ) ) )
5 oeordi 6821 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( B  e.  A  ->  ( C  ^o  B
)  e.  ( C  ^o  A ) ) )
653adant2 976 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( B  e.  A  ->  ( C  ^o  B )  e.  ( C  ^o  A ) ) )
74, 6orim12d 812 . . . 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 3461 . . . . . 6  |-  ( C  e.  ( On  \  2o )  ->  C  e.  On )
1093ad2ant3 980 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  C  e.  On )
11 simp1 957 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  A  e.  On )
12 oecl 6772 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
1310, 11, 12syl2anc 643 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  A )  e.  On )
14 simp2 958 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  B  e.  On )
15 oecl 6772 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1610, 14, 15syl2anc 643 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  B )  e.  On )
17 eloni 4583 . . . . 5  |-  ( ( C  ^o  A )  e.  On  ->  Ord  ( C  ^o  A ) )
18 eloni 4583 . . . . 5  |-  ( ( C  ^o  B )  e.  On  ->  Ord  ( C  ^o  B ) )
19 ordtri2 4608 . . . . 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 464 . . . 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 643 . . 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 4583 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
23 eloni 4583 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
24 ordtri2 4608 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2522, 23, 24syl2an 464 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A
) ) )
26253adant3 977 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
278, 21, 263imtr4d 260 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  e.  ( C  ^o  B )  ->  A  e.  B )
)
282, 27impbid 184 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 3    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1652    e. wcel 1725    \ cdif 3309   Ord word 4572   Oncon0 4573  (class class class)co 6072   2oc2o 6709    ^o coe 6714
This theorem is referenced by:  oeword  6824  oeeui  6836  omabs  6881  cantnflem3  7636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-omul 6720  df-oexp 6721
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