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Theorem oacan 6548
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 6547 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On  /\  A  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
213comr 1159 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
3 oaord 6547 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
433com13 1156 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
52, 4orbi12d 690 . . 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 285 . 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 4404 . . . 4  |-  ( B  e.  On  ->  Ord  B )
8 eloni 4404 . . . 4  |-  ( C  e.  On  ->  Ord  C )
9 ordtri3 4430 . . . 4  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
107, 8, 9syl2an 463 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
11103adant1 973 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
12 oacl 6536 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
13 eloni 4404 . . . . 5  |-  ( ( A  +o  B )  e.  On  ->  Ord  ( A  +o  B
) )
1412, 13syl 15 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  +o  B ) )
15 oacl 6536 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  +o  C
)  e.  On )
16 eloni 4404 . . . . 5  |-  ( ( A  +o  C )  e.  On  ->  Ord  ( A  +o  C
) )
1715, 16syl 15 . . . 4  |-  ( ( A  e.  On  /\  C  e.  On )  ->  Ord  ( A  +o  C ) )
18 ordtri3 4430 . . . 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 463 . . 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 1237 . 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 277 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   Ord word 4393   Oncon0 4394  (class class class)co 5860    +o coa 6478
This theorem is referenced by:  oawordeulem  6554
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-oadd 6485
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