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Theorem oaord 7208
Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)
Assertion
Ref Expression
oaord  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  ( C  +o  A )  e.  ( C  +o  B ) ) )

Proof of Theorem oaord
StepHypRef Expression
1 oaordi 7207 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A
)  e.  ( C  +o  B ) ) )
213adant1 1014 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A )  e.  ( C  +o  B ) ) )
3 oveq2 6303 . . . . . 6  |-  ( A  =  B  ->  ( C  +o  A )  =  ( C  +o  B
) )
43a1i 11 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  =  B  ->  ( C  +o  A )  =  ( C  +o  B ) ) )
5 oaordi 7207 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B
)  e.  ( C  +o  A ) ) )
653adant2 1015 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B )  e.  ( C  +o  A ) ) )
74, 6orim12d 836 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  =  B  \/  B  e.  A
)  ->  ( ( C  +o  A )  =  ( C  +o  B
)  \/  ( C  +o  B )  e.  ( C  +o  A
) ) ) )
87con3d 133 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  ( ( C  +o  A )  =  ( C  +o  B )  \/  ( C  +o  B )  e.  ( C  +o  A ) )  ->  -.  ( A  =  B  \/  B  e.  A )
) )
9 df-3an 975 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) )
10 ancom 450 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) 
<->  ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) ) )
11 anandi 826 . . . . . 6  |-  ( ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) )  <->  ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) ) )
129, 10, 113bitri 271 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) ) )
13 oacl 7197 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
14 eloni 4894 . . . . . . 7  |-  ( ( C  +o  A )  e.  On  ->  Ord  ( C  +o  A
) )
1513, 14syl 16 . . . . . 6  |-  ( ( C  e.  On  /\  A  e.  On )  ->  Ord  ( C  +o  A ) )
16 oacl 7197 . . . . . . 7  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
17 eloni 4894 . . . . . . 7  |-  ( ( C  +o  B )  e.  On  ->  Ord  ( C  +o  B
) )
1816, 17syl 16 . . . . . 6  |-  ( ( C  e.  On  /\  B  e.  On )  ->  Ord  ( C  +o  B ) )
1915, 18anim12i 566 . . . . 5  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) )  -> 
( Ord  ( C  +o  A )  /\  Ord  ( C  +o  B
) ) )
2012, 19sylbi 195 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  ( C  +o  A
)  /\  Ord  ( C  +o  B ) ) )
21 ordtri2 4919 . . . 4  |-  ( ( Ord  ( C  +o  A )  /\  Ord  ( C  +o  B
) )  ->  (
( C  +o  A
)  e.  ( C  +o  B )  <->  -.  (
( C  +o  A
)  =  ( C  +o  B )  \/  ( C  +o  B
)  e.  ( C  +o  A ) ) ) )
2220, 21syl 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  +o  A
)  e.  ( C  +o  B )  <->  -.  (
( C  +o  A
)  =  ( C  +o  B )  \/  ( C  +o  B
)  e.  ( C  +o  A ) ) ) )
23 eloni 4894 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
24 eloni 4894 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
2523, 24anim12i 566 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( Ord  A  /\  Ord  B ) )
26253adant3 1016 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  A  /\  Ord  B
) )
27 ordtri2 4919 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2826, 27syl 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
298, 22, 283imtr4d 268 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  +o  A
)  e.  ( C  +o  B )  ->  A  e.  B )
)
302, 29impbid 191 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  ( C  +o  A )  e.  ( C  +o  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   Ord word 4883   Oncon0 4884  (class class class)co 6295    +o coa 7139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-oadd 7146
This theorem is referenced by:  oacan  7209  oaword  7210  oaord1  7212  oa00  7220  oalimcl  7221  oaass  7222  odi  7240  oneo  7242  omeulem1  7243  omeulem2  7244  oeeui  7263  omxpenlem  7630  cantnflt  8103  cantnflem1d  8119  cantnflem1  8120  cantnfltOLD  8133  cantnflem1dOLD  8142  cantnflem1OLD  8143
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