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Theorem oacan 6758
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 6757 . . . . 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 6757 . . . . 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 4559 . . . 4  |-  ( B  e.  On  ->  Ord  B )
8 eloni 4559 . . . 4  |-  ( C  e.  On  ->  Ord  C )
9 ordtri3 4585 . . . 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 6746 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
13 eloni 4559 . . . . 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 6746 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  +o  C
)  e.  On )
16 eloni 4559 . . . . 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 4585 . . . 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 1721   Ord word 4548   Oncon0 4549  (class class class)co 6048    +o coa 6688
This theorem is referenced by:  oawordeulem  6764
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-oadd 6695
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