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

Step	Hyp	Ref	Expression
1		oaord 6991	. . . 4 |- ( ( B e. On /\ A e. On /\ C e. On ) -> ( B e. A <-> ( C +o B ) e. ( C +o A ) ) )
2	1	3com12 1191	. . 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 4758	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> -. B e. A ) )
5	4	3adant3 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 )
7	6	ancoms 453	. . . 4 |- ( ( A e. On /\ C e. On ) -> ( C +o A ) e. On )
8	7	3adant2 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 )
10	9	ancoms 453	. . . 4 |- ( ( B e. On /\ C e. On ) -> ( C +o B ) e. On )
11	10	3adant1 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 ) ) )
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 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