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Theorem oaword 6589
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 6587 . . . 4  |-  ( ( B  e.  On  /\  A  e.  On  /\  C  e.  On )  ->  ( B  e.  A  <->  ( C  +o  B )  e.  ( C  +o  A ) ) )
213com12 1155 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  A  <->  ( C  +o  B )  e.  ( C  +o  A ) ) )
32notbid 285 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  B  e.  A  <->  -.  ( C  +o  B
)  e.  ( C  +o  A ) ) )
4 ontri1 4463 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
543adant3 975 . 2  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  <->  -.  B  e.  A ) )
6 oacl 6576 . . . . 5  |-  ( ( C  e.  On  /\  A  e.  On )  ->  ( C  +o  A
)  e.  On )
76ancoms 439 . . . 4  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( C  +o  A
)  e.  On )
873adant2 974 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  +o  A )  e.  On )
9 oacl 6576 . . . . 5  |-  ( ( C  e.  On  /\  B  e.  On )  ->  ( C  +o  B
)  e.  On )
109ancoms 439 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( C  +o  B
)  e.  On )
11103adant1 973 . . 3  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( C  +o  B )  e.  On )
12 ontri1 4463 . . 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 642 . 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 276 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 176    /\ w3a 934    e. wcel 1701    C_ wss 3186   Oncon0 4429  (class class class)co 5900    +o coa 6518
This theorem is referenced by:  oaword1  6592  oaass  6601  omwordri  6612  omlimcl  6618  oaabs2  6685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-recs 6430  df-rdg 6465  df-oadd 6525
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