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Theorem oaord 6792
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 6791 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A
)  e.  ( C  +o  B ) ) )
213adant1 976 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A )  e.  ( C  +o  B ) ) )
3 oveq2 6091 . . . . . 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 6791 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B
)  e.  ( C  +o  A ) ) )
653adant2 977 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B )  e.  ( C  +o  A ) ) )
74, 6orim12d 813 . . . 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 128 . . 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 939 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) )
10 ancom 439 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) 
<->  ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) ) )
11 anandi 803 . . . . . 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 264 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) ) )
13 oacl 6781 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
14 eloni 4593 . . . . . . 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 6781 . . . . . . 7  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
17 eloni 4593 . . . . . . 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 551 . . . . 5  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) )  -> 
( Ord  ( C  +o  A )  /\  Ord  ( C  +o  B
) ) )
2012, 19sylbi 189 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  ( C  +o  A
)  /\  Ord  ( C  +o  B ) ) )
21 ordtri2 4618 . . . 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 4593 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
24 eloni 4593 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
2523, 24anim12i 551 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( Ord  A  /\  Ord  B ) )
26253adant3 978 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  A  /\  Ord  B
) )
27 ordtri2 4618 . . . 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 261 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  +o  A
)  e.  ( C  +o  B )  ->  A  e.  B )
)
302, 29impbid 185 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 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   Ord word 4582   Oncon0 4583  (class class class)co 6083    +o coa 6723
This theorem is referenced by:  oacan  6793  oaword  6794  oaord1  6796  oa00  6804  oalimcl  6805  oaass  6806  odi  6824  oneo  6826  omeulem1  6827  omeulem2  6828  oeeui  6847  omxpenlem  7211  cantnflt  7629  cantnflem1d  7646  cantnflem1  7647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-oadd 6730
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