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Theorem oecan 6770
Description: Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oecan  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  <->  B  =  C ) )

Proof of Theorem oecan
StepHypRef Expression
1 oeordi 6768 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  ( On  \  2o ) )  -> 
( B  e.  C  ->  ( A  ^o  B
)  e.  ( A  ^o  C ) ) )
21ancoms 440 . . . . . 6  |-  ( ( A  e.  ( On 
\  2o )  /\  C  e.  On )  ->  ( B  e.  C  ->  ( A  ^o  B
)  e.  ( A  ^o  C ) ) )
323adant2 976 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  C  ->  ( A  ^o  B )  e.  ( A  ^o  C ) ) )
4 oeordi 6768 . . . . . . 7  |-  ( ( B  e.  On  /\  A  e.  ( On  \  2o ) )  -> 
( C  e.  B  ->  ( A  ^o  C
)  e.  ( A  ^o  B ) ) )
54ancoms 440 . . . . . 6  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On )  ->  ( C  e.  B  ->  ( A  ^o  C
)  e.  ( A  ^o  B ) ) )
653adant3 977 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( C  e.  B  ->  ( A  ^o  C )  e.  ( A  ^o  B ) ) )
73, 6orim12d 812 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( B  e.  C  \/  C  e.  B
)  ->  ( ( A  ^o  B )  e.  ( A  ^o  C
)  \/  ( A  ^o  C )  e.  ( A  ^o  B
) ) ) )
87con3d 127 . . 3  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  ( ( A  ^o  B )  e.  ( A  ^o  C )  \/  ( A  ^o  C )  e.  ( A  ^o  B ) )  ->  -.  ( B  e.  C  \/  C  e.  B )
) )
9 eldifi 3414 . . . . . 6  |-  ( A  e.  ( On  \  2o )  ->  A  e.  On )
1093ad2ant1 978 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  A  e.  On )
11 simp2 958 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  B  e.  On )
12 oecl 6719 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
1310, 11, 12syl2anc 643 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  B )  e.  On )
14 simp3 959 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  C  e.  On )
15 oecl 6719 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  ^o  C
)  e.  On )
1610, 14, 15syl2anc 643 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  C )  e.  On )
17 eloni 4534 . . . . 5  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
18 eloni 4534 . . . . 5  |-  ( ( A  ^o  C )  e.  On  ->  Ord  ( A  ^o  C ) )
19 ordtri3 4560 . . . . 5  |-  ( ( Ord  ( A  ^o  B )  /\  Ord  ( A  ^o  C ) )  ->  ( ( A  ^o  B )  =  ( A  ^o  C
)  <->  -.  ( ( A  ^o  B )  e.  ( A  ^o  C
)  \/  ( A  ^o  C )  e.  ( A  ^o  B
) ) ) )
2017, 18, 19syl2an 464 . . . 4  |-  ( ( ( A  ^o  B
)  e.  On  /\  ( A  ^o  C )  e.  On )  -> 
( ( A  ^o  B )  =  ( A  ^o  C )  <->  -.  ( ( A  ^o  B )  e.  ( A  ^o  C )  \/  ( A  ^o  C )  e.  ( A  ^o  B ) ) ) )
2113, 16, 20syl2anc 643 . . 3  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  <->  -.  (
( A  ^o  B
)  e.  ( A  ^o  C )  \/  ( A  ^o  C
)  e.  ( A  ^o  B ) ) ) )
22 eloni 4534 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
23 eloni 4534 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
24 ordtri3 4560 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2522, 23, 24syl2an 464 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
26253adant1 975 . . 3  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
278, 21, 263imtr4d 260 . 2  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  ->  B  =  C )
)
28 oveq2 6030 . 2  |-  ( B  =  C  ->  ( A  ^o  B )  =  ( A  ^o  C
) )
2927, 28impbid1 195 1  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  <->  B  =  C ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1717    \ cdif 3262   Ord word 4523   Oncon0 4524  (class class class)co 6022   2oc2o 6656    ^o coe 6661
This theorem is referenced by:  oeword  6771  infxpenc2lem1  7835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-omul 6667  df-oexp 6668
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