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Theorem oacan 7215
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 7214 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On  /\  A  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
213comr 1204 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
3 oaord 7214 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
433com13 1201 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
52, 4orbi12d 709 . . 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 294 . 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 4897 . . . 4  |-  ( B  e.  On  ->  Ord  B )
8 eloni 4897 . . . 4  |-  ( C  e.  On  ->  Ord  C )
9 ordtri3 4923 . . . 4  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
107, 8, 9syl2an 477 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
11103adant1 1014 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
12 oacl 7203 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
13 eloni 4897 . . . . 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 7203 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  +o  C
)  e.  On )
16 eloni 4897 . . . . 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 4923 . . . 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 477 . . 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 1283 . 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 286 1  |-  ( ( A  e.  On  /\  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    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   Ord word 4886   Oncon0 4887  (class class class)co 6296    +o coa 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-oadd 7152
This theorem is referenced by:  oawordeulem  7221
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