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Theorem oecan 7235
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 7233 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  ( On  \  2o ) )  -> 
( B  e.  C  ->  ( A  ^o  B
)  e.  ( A  ^o  C ) ) )
21ancoms 453 . . . . . 6  |-  ( ( A  e.  ( On 
\  2o )  /\  C  e.  On )  ->  ( B  e.  C  ->  ( A  ^o  B
)  e.  ( A  ^o  C ) ) )
323adant2 1015 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  C  ->  ( A  ^o  B )  e.  ( A  ^o  C ) ) )
4 oeordi 7233 . . . . . . 7  |-  ( ( B  e.  On  /\  A  e.  ( On  \  2o ) )  -> 
( C  e.  B  ->  ( A  ^o  C
)  e.  ( A  ^o  B ) ) )
54ancoms 453 . . . . . 6  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On )  ->  ( C  e.  B  ->  ( A  ^o  C
)  e.  ( A  ^o  B ) ) )
653adant3 1016 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( C  e.  B  ->  ( A  ^o  C )  e.  ( A  ^o  B ) ) )
73, 6orim12d 836 . . . 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 133 . . 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 3626 . . . . . 6  |-  ( A  e.  ( On  \  2o )  ->  A  e.  On )
1093ad2ant1 1017 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  A  e.  On )
11 simp2 997 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  B  e.  On )
12 oecl 7184 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
1310, 11, 12syl2anc 661 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  B )  e.  On )
14 simp3 998 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  C  e.  On )
15 oecl 7184 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  ^o  C
)  e.  On )
1610, 14, 15syl2anc 661 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  C )  e.  On )
17 eloni 4888 . . . . 5  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
18 eloni 4888 . . . . 5  |-  ( ( A  ^o  C )  e.  On  ->  Ord  ( A  ^o  C ) )
19 ordtri3 4914 . . . . 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 477 . . . 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 661 . . 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 4888 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
23 eloni 4888 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
24 ordtri3 4914 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2522, 23, 24syl2an 477 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
26253adant1 1014 . . 3  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
278, 21, 263imtr4d 268 . 2  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  ->  B  =  C )
)
28 oveq2 6290 . 2  |-  ( B  =  C  ->  ( A  ^o  B )  =  ( A  ^o  C
) )
2927, 28impbid1 203 1  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  <->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ w3a 973    = wceq 1379    e. wcel 1767    \ cdif 3473   Ord word 4877   Oncon0 4878  (class class class)co 6282   2oc2o 7121    ^o coe 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-oexp 7133
This theorem is referenced by:  oeword  7236  infxpenc2lem1  8392
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