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Theorem oaword 6793
Description: Weak ordering property of ordinal addition. (Contributed by NM, 6-Dec-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
oaword  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  ( C  +o  A )  C_  ( C  +o  B ) ) )

Proof of Theorem oaword
StepHypRef Expression
1 oaord 6791 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On  /\  C  e.  On )  ->  ( B  e.  A  <->  ( C  +o  B )  e.  ( C  +o  A ) ) )
213com12 1158 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  A  <->  ( C  +o  B )  e.  ( C  +o  A ) ) )
32notbid 287 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  B  e.  A  <->  -.  ( C  +o  B
)  e.  ( C  +o  A ) ) )
4 ontri1 4616 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
543adant3 978 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
6 oacl 6780 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
76ancoms 441 . . . 4  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( C  +o  A
)  e.  On )
873adant2 977 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  +o  A )  e.  On )
9 oacl 6780 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
109ancoms 441 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( C  +o  B
)  e.  On )
11103adant1 976 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  +o  B )  e.  On )
12 ontri1 4616 . . 3  |-  ( ( ( C  +o  A
)  e.  On  /\  ( C  +o  B
)  e.  On )  ->  ( ( C  +o  A )  C_  ( C  +o  B
)  <->  -.  ( C  +o  B )  e.  ( C  +o  A ) ) )
138, 11, 12syl2anc 644 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( C  +o  A
)  C_  ( C  +o  B )  <->  -.  ( C  +o  B )  e.  ( C  +o  A
) ) )
143, 5, 133bitr4d 278 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  ( C  +o  A )  C_  ( C  +o  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ w3a 937    e. wcel 1726    C_ wss 3321   Oncon0 4582  (class class class)co 6082    +o coa 6722
This theorem is referenced by:  oaword1  6796  oaass  6805  omwordri  6816  omlimcl  6822  oaabs2  6889
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-recs 6634  df-rdg 6669  df-oadd 6729
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