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Theorem oecan 6824
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 6822 . . . . . . 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 6822 . . . . . . 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 3461 . . . . . 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 6773 . . . . 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 6773 . . . . 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 4583 . . . . 5  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
18 eloni 4583 . . . . 5  |-  ( ( A  ^o  C )  e.  On  ->  Ord  ( A  ^o  C ) )
19 ordtri3 4609 . . . . 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 4583 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
23 eloni 4583 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
24 ordtri3 4609 . . . . 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 6081 . 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 1652    e. wcel 1725    \ cdif 3309   Ord word 4572   Oncon0 4573  (class class class)co 6073   2oc2o 6710    ^o coe 6715
This theorem is referenced by:  oeword  6825  infxpenc2lem1  7892
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 4693
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 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-oexp 6722
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