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Theorem oacan 7249
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 7248 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On  /\  A  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
213comr 1216 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  C  <->  ( A  +o  B )  e.  ( A  +o  C ) ) )
3 oaord 7248 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On  /\  A  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
433com13 1213 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  e.  B  <->  ( A  +o  C )  e.  ( A  +o  B ) ) )
52, 4orbi12d 716 . . 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 296 . 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 5433 . . . 4  |-  ( B  e.  On  ->  Ord  B )
8 eloni 5433 . . . 4  |-  ( C  e.  On  ->  Ord  C )
9 ordtri3 5459 . . . 4  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
107, 8, 9syl2an 480 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
11103adant1 1026 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
12 oacl 7237 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
13 eloni 5433 . . . . 5  |-  ( ( A  +o  B )  e.  On  ->  Ord  ( A  +o  B
) )
1412, 13syl 17 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  Ord  ( A  +o  B ) )
15 oacl 7237 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  +o  C
)  e.  On )
16 eloni 5433 . . . . 5  |-  ( ( A  +o  C )  e.  On  ->  Ord  ( A  +o  C
) )
1715, 16syl 17 . . . 4  |-  ( ( A  e.  On  /\  C  e.  On )  ->  Ord  ( A  +o  C ) )
18 ordtri3 5459 . . . 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 480 . . 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 1323 . 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 290 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 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   Ord word 5422   Oncon0 5423  (class class class)co 6290    +o coa 7179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-oadd 7186
This theorem is referenced by:  oawordeulem  7255
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