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Theorem oaord 6541
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 6540 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A
)  e.  ( C  +o  B ) ) )
213adant1 973 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  ->  ( C  +o  A )  e.  ( C  +o  B ) ) )
3 oveq2 5828 . . . . . 6  |-  ( A  =  B  ->  ( C  +o  A )  =  ( C  +o  B
) )
43a1i 10 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  =  B  ->  ( C  +o  A )  =  ( C  +o  B ) ) )
5 oaordi 6540 . . . . . 6  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B
)  e.  ( C  +o  A ) ) )
653adant2 974 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  A  ->  ( C  +o  B )  e.  ( C  +o  A ) ) )
74, 6orim12d 811 . . . 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 125 . . 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 936 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) )
10 ancom 437 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  C  e.  On ) 
<->  ( C  e.  On  /\  ( A  e.  On  /\  B  e.  On ) ) )
11 anandi 801 . . . . . 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 262 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  <->  ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) ) )
13 oacl 6530 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
14 eloni 4401 . . . . . . 7  |-  ( ( C  +o  A )  e.  On  ->  Ord  ( C  +o  A
) )
1513, 14syl 15 . . . . . 6  |-  ( ( C  e.  On  /\  A  e.  On )  ->  Ord  ( C  +o  A ) )
16 oacl 6530 . . . . . . 7  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
17 eloni 4401 . . . . . . 7  |-  ( ( C  +o  B )  e.  On  ->  Ord  ( C  +o  B
) )
1816, 17syl 15 . . . . . 6  |-  ( ( C  e.  On  /\  B  e.  On )  ->  Ord  ( C  +o  B ) )
1915, 18anim12i 549 . . . . 5  |-  ( ( ( C  e.  On  /\  A  e.  On )  /\  ( C  e.  On  /\  B  e.  On ) )  -> 
( Ord  ( C  +o  A )  /\  Ord  ( C  +o  B
) ) )
2012, 19sylbi 187 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  ( C  +o  A
)  /\  Ord  ( C  +o  B ) ) )
21 ordtri2 4426 . . . 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 15 . . 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 4401 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
24 eloni 4401 . . . . . 6  |-  ( B  e.  On  ->  Ord  B )
2523, 24anim12i 549 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( Ord  A  /\  Ord  B ) )
26253adant3 975 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( Ord  A  /\  Ord  B
) )
27 ordtri2 4426 . . . 4  |-  ( ( Ord  A  /\  Ord  B )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A ) ) )
2826, 27syl 15 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  e.  B  <->  -.  ( A  =  B  \/  B  e.  A )
) )
298, 22, 283imtr4d 259 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  +o  A
)  e.  ( C  +o  B )  ->  A  e.  B )
)
302, 29impbid 183 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   Ord word 4390   Oncon0 4391  (class class class)co 5820    +o coa 6472
This theorem is referenced by:  oacan  6542  oaword  6543  oaord1  6545  oa00  6553  oalimcl  6554  oaass  6555  odi  6573  oneo  6575  omeulem1  6576  omeulem2  6577  oeeui  6596  omxpenlem  6959  cantnflt  7369  cantnflem1d  7386  cantnflem1  7387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-recs 6384  df-rdg 6419  df-oadd 6479
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