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Theorem oeord 7227
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 7226 . . 3  |-  ( ( B  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( A  e.  B  ->  ( C  ^o  A
)  e.  ( C  ^o  B ) ) )
213adant1 1009 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  ->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
3 oveq2 6283 . . . . . 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 7226 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( B  e.  A  ->  ( C  ^o  B
)  e.  ( C  ^o  A ) ) )
653adant2 1010 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( B  e.  A  ->  ( C  ^o  B )  e.  ( C  ^o  A ) ) )
74, 6orim12d 835 . . . 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 133 . . 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 3619 . . . . . 6  |-  ( C  e.  ( On  \  2o )  ->  C  e.  On )
1093ad2ant3 1014 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  C  e.  On )
11 simp1 991 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  A  e.  On )
12 oecl 7177 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
1310, 11, 12syl2anc 661 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  A )  e.  On )
14 simp2 992 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  B  e.  On )
15 oecl 7177 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1610, 14, 15syl2anc 661 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  B )  e.  On )
17 eloni 4881 . . . . 5  |-  ( ( C  ^o  A )  e.  On  ->  Ord  ( C  ^o  A ) )
18 eloni 4881 . . . . 5  |-  ( ( C  ^o  B )  e.  On  ->  Ord  ( C  ^o  B ) )
19 ordtri2 4906 . . . . 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 477 . . . 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 661 . . 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 4881 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
23 eloni 4881 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
24 ordtri2 4906 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2522, 23, 24syl2an 477 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A
) ) )
26253adant3 1011 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
278, 21, 263imtr4d 268 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  e.  ( C  ^o  B )  ->  A  e.  B )
)
282, 27impbid 191 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ w3a 968    = wceq 1374    e. wcel 1762    \ cdif 3466   Ord word 4870   Oncon0 4871  (class class class)co 6275   2oc2o 7114    ^o coe 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-oexp 7126
This theorem is referenced by:  oeword  7229  oeeui  7241  omabs  7286  cantnflem3  8099  cantnflem3OLD  8121
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