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Theorem oaword 6993
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 6991 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On  /\  C  e.  On )  ->  ( B  e.  A  <->  ( C  +o  B )  e.  ( C  +o  A ) ) )
213com12 1191 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  A  <->  ( C  +o  B )  e.  ( C  +o  A ) ) )
32notbid 294 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  B  e.  A  <->  -.  ( C  +o  B
)  e.  ( C  +o  A ) ) )
4 ontri1 4758 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
543adant3 1008 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
6 oacl 6980 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
76ancoms 453 . . . 4  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( C  +o  A
)  e.  On )
873adant2 1007 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  +o  A )  e.  On )
9 oacl 6980 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
109ancoms 453 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( C  +o  B
)  e.  On )
11103adant1 1006 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  +o  B )  e.  On )
12 ontri1 4758 . . 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 661 . 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 285 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  ( C  +o  A )  C_  ( C  +o  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 965    e. wcel 1756    C_ wss 3333   Oncon0 4724  (class class class)co 6096    +o coa 6922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-oadd 6929
This theorem is referenced by:  oaword1  6996  oaass  7005  omwordri  7016  omlimcl  7022  oaabs2  7089
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