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Theorem oaord 6547
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 6546 . . 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 5868 . . . . . 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 6546 . . . . . 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 6536 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
14 eloni 4404 . . . . . . 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 6536 . . . . . . 7  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
17 eloni 4404 . . . . . . 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 4429 . . . 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 4404 . . . . . 6  |-  ( A  e.  On  ->  Ord  A )
24 eloni 4404 . . . . . 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 4429 . . . 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 1625    e. wcel 1686   Ord word 4393   Oncon0 4394  (class class class)co 5860    +o coa 6478
This theorem is referenced by:  oacan  6548  oaword  6549  oaord1  6551  oa00  6559  oalimcl  6560  oaass  6561  odi  6579  oneo  6581  omeulem1  6582  omeulem2  6583  oeeui  6602  omxpenlem  6965  cantnflt  7375  cantnflem1d  7392  cantnflem1  7393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-oadd 6485
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