Theorem oaword 7198

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 7196	. . . 4 |- ( ( B e. On /\ A e. On /\ C e. On ) -> ( B e. A <-> ( C +o B ) e. ( C +o A ) ) )
2	1	3com12 1200	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( B e. A <-> ( C +o B ) e. ( C +o A ) ) )
3	2	notbid 294	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( -. B e. A <-> -. ( C +o B ) e. ( C +o A ) ) )
4		ontri1 4912	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> -. B e. A ) )
5	4	3adant3 1016	. 2 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B <-> -. B e. A ) )
6		oacl 7185	. . . . 5 |- ( ( C e. On /\ A e. On ) -> ( C +o A ) e. On )
7	6	ancoms 453	. . . 4 |- ( ( A e. On /\ C e. On ) -> ( C +o A ) e. On )
8	7	3adant2 1015	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( C +o A ) e. On )
9		oacl 7185	. . . . 5 |- ( ( C e. On /\ B e. On ) -> ( C +o B ) e. On )
10	9	ancoms 453	. . . 4 |- ( ( B e. On /\ C e. On ) -> ( C +o B ) e. On )
11	10	3adant1 1014	. . 3 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( C +o B ) e. On )
12		ontri1 4912	. . 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 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 ) ) )
14	3, 5, 13	3bitr4d 285	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 184   /\ w3a 973   e. wcel 1767   C_ wss 3476   Oncon0 4878  (class class class)co 6284   +o coa 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-oadd 7134

This theorem is referenced by:  oaword1  7201  oaass  7210  omwordri  7221  omlimcl  7227  oaabs2  7294