Theorem oaword 7250

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

Step	Hyp	Ref	Expression
1		oaord 7248	. . . 4 |- ( ( B e. On /\ A e. On /\ C e. On ) -> ( B e. A <-> ( C +o B ) e. ( C +o A ) ) )
2	1	3com12 1212	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( B e. A <-> ( C +o B ) e. ( C +o A ) ) )
3	2	notbid 296	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( -. B e. A <-> -. ( C +o B ) e. ( C +o A ) ) )
4		ontri1 5457	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> -. B e. A ) )
5	4	3adant3 1028	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> -. B e. A ) )
6		oacl 7237	. . . . 5 |- ( ( C e. On /\ A e. On ) -> ( C +o A ) e. On )
7	6	ancoms 455	. . . 4 |- ( ( A e. On /\ C e. On ) -> ( C +o A ) e. On )
8	7	3adant2 1027	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( C +o A ) e. On )
9		oacl 7237	. . . . 5 |- ( ( C e. On /\ B e. On ) -> ( C +o B ) e. On )
10	9	ancoms 455	. . . 4 |- ( ( B e. On /\ C e. On ) -> ( C +o B ) e. On )
11	10	3adant1 1026	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( C +o B ) e. On )
12		ontri1 5457	. . 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 ) ) )
13	8, 11, 12	syl2anc 667	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( ( C +o A ) C_ ( C +o B ) <-> -. ( C +o B ) e. ( C +o A ) ) )
14	3, 5, 13	3bitr4d 289	1 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> ( C +o A ) C_ ( C +o B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ w3a 985   e. wcel 1887   C_ wss 3404   Oncon0 5423  (class class class)co 6290   +o coa 7179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-oadd 7186

This theorem is referenced by:  oaword1  7253  oaass  7262  omwordri  7273  omlimcl  7279  oaabs2  7346