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Theorem oaord 6782
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 6781 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A
)  e.  ( C  +o  B ) ) )
213adant1 975 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A )  e.  ( C  +o  B ) ) )
3 oveq2 6081 . . . . . 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 6781 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B
)  e.  ( C  +o  A ) ) )
653adant2 976 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B )  e.  ( C  +o  A ) ) )
74, 6orim12d 812 . . . 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 127 . . 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 938 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) )
10 ancom 438 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) 
<->  ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) ) )
11 anandi 802 . . . . . 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 263 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) ) )
13 oacl 6771 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
14 eloni 4583 . . . . . . 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 6771 . . . . . . 7  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
17 eloni 4583 . . . . . . 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 550 . . . . 5  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) )  -> 
( Ord  ( C  +o  A )  /\  Ord  ( C  +o  B
) ) )
2012, 19sylbi 188 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  ( C  +o  A
)  /\  Ord  ( C  +o  B ) ) )
21 ordtri2 4608 . . . 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 4583 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
24 eloni 4583 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
2523, 24anim12i 550 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( Ord  A  /\  Ord  B ) )
26253adant3 977 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  A  /\  Ord  B
) )
27 ordtri2 4608 . . . 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 260 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  +o  A
)  e.  ( C  +o  B )  ->  A  e.  B )
)
302, 29impbid 184 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  ( C  +o  A )  e.  ( C  +o  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   Ord word 4572   Oncon0 4573  (class class class)co 6073    +o coa 6713
This theorem is referenced by:  oacan  6783  oaword  6784  oaord1  6786  oa00  6794  oalimcl  6795  oaass  6796  odi  6814  oneo  6816  omeulem1  6817  omeulem2  6818  oeeui  6837  omxpenlem  7201  cantnflt  7619  cantnflem1d  7636  cantnflem1  7637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-oadd 6720
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