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Theorem oeord 6519
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 6518 . . 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 5765 . . . . . 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 6518 . . . . . 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 3240 . . . . . 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 6469 . . . . 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 6469 . . . . 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 4339 . . . . 5  |-  ( ( C  ^o  A )  e.  On  ->  Ord  ( C  ^o  A ) )
18 eloni 4339 . . . . 5  |-  ( ( C  ^o  B )  e.  On  ->  Ord  ( C  ^o  B ) )
19 ordtri2 4364 . . . . 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 4339 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
23 eloni 4339 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
24 ordtri2 4364 . . . . 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 3091   Ord word 4328   Oncon0 4329  (class class class)co 5757   2oc2o 6406    ^o coe 6411
This theorem is referenced by:  oeword  6521  oeeui  6533  omabs  6578  cantnflem3  7326
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 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-oexp 6418
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