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Theorem oeord 6586
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 6585 . . 3  |-  ( ( B  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( A  e.  B  ->  ( C  ^o  A
)  e.  ( C  ^o  B ) ) )
213adant1 973 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  ->  ( C  ^o  A )  e.  ( C  ^o  B ) ) )
3 oveq2 5866 . . . . . 6  |-  ( A  =  B  ->  ( C  ^o  A )  =  ( C  ^o  B
) )
43a1i 10 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  =  B  ->  ( C  ^o  A )  =  ( C  ^o  B ) ) )
5 oeordi 6585 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  ( On  \  2o ) )  -> 
( B  e.  A  ->  ( C  ^o  B
)  e.  ( C  ^o  A ) ) )
653adant2 974 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( B  e.  A  ->  ( C  ^o  B )  e.  ( C  ^o  A ) ) )
74, 6orim12d 811 . . . 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 125 . . 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 3298 . . . . . 6  |-  ( C  e.  ( On  \  2o )  ->  C  e.  On )
1093ad2ant3 978 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  C  e.  On )
11 simp1 955 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  A  e.  On )
12 oecl 6536 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  ^o  A
)  e.  On )
1310, 11, 12syl2anc 642 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  A )  e.  On )
14 simp2 956 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  B  e.  On )
15 oecl 6536 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  ^o  B
)  e.  On )
1610, 14, 15syl2anc 642 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( C  ^o  B )  e.  On )
17 eloni 4402 . . . . 5  |-  ( ( C  ^o  A )  e.  On  ->  Ord  ( C  ^o  A ) )
18 eloni 4402 . . . . 5  |-  ( ( C  ^o  B )  e.  On  ->  Ord  ( C  ^o  B ) )
19 ordtri2 4427 . . . . 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 463 . . . 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 642 . . 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 4402 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
23 eloni 4402 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
24 ordtri2 4427 . . . . 5  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2522, 23, 24syl2an 463 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A
) ) )
26253adant3 975 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
278, 21, 263imtr4d 259 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  ( On  \  2o ) )  ->  (
( C  ^o  A
)  e.  ( C  ^o  B )  ->  A  e.  B )
)
282, 27impbid 183 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 176    \/ wo 357    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149   Ord word 4391   Oncon0 4392  (class class class)co 5858   2oc2o 6473    ^o coe 6478
This theorem is referenced by:  oeword  6588  oeeui  6600  omabs  6645  cantnflem3  7393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485
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