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Theorem oacan 6721
Description: Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58. (Contributed by NM, 5-Dec-2004.)
Assertion
Ref Expression
oacan  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  +o  B
)  =  ( A  +o  C )  <->  B  =  C ) )

Proof of Theorem oacan
StepHypRef Expression
1 oaord 6720 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On  /\  A  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
213comr 1161 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
3 oaord 6720 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
433com13 1158 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
52, 4orbi12d 691 . . 3  |-  ( ( A  e.  On  /\  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
) ) ) )
65notbid 286 . 2  |-  ( ( A  e.  On  /\  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
) ) ) )
7 eloni 4526 . . . 4  |-  ( B  e.  On  ->  Ord  B )
8 eloni 4526 . . . 4  |-  ( C  e.  On  ->  Ord  C )
9 ordtri3 4552 . . . 4  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
107, 8, 9syl2an 464 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
11103adant1 975 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
12 oacl 6709 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
13 eloni 4526 . . . . 5  |-  ( ( A  +o  B )  e.  On  ->  Ord  ( A  +o  B
) )
1412, 13syl 16 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  +o  B ) )
15 oacl 6709 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  +o  C
)  e.  On )
16 eloni 4526 . . . . 5  |-  ( ( A  +o  C )  e.  On  ->  Ord  ( A  +o  C
) )
1715, 16syl 16 . . . 4  |-  ( ( A  e.  On  /\  C  e.  On )  ->  Ord  ( A  +o  C ) )
18 ordtri3 4552 . . . 4  |-  ( ( 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 ) ) ) )
1914, 17, 18syl2an 464 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  ( A  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 ) ) ) )
20193impdi 1239 . 2  |-  ( ( A  e.  On  /\  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 ) ) ) )
216, 11, 203bitr4rd 278 1  |-  ( ( A  e.  On  /\  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    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   Ord word 4515   Oncon0 4516  (class class class)co 6014    +o coa 6651
This theorem is referenced by:  oawordeulem  6727
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 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-recs 6563  df-rdg 6598  df-oadd 6658
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