Theorem oaword 7216

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 7214	. . . 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 4921	. . 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 7203	. . . . 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 7203	. . . . 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 4921	. . 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 1819   C_ wss 3471   Oncon0 4887  (class class class)co 6296   +o coa 7145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-oadd 7152

This theorem is referenced by:  oaword1  7219  oaass  7228  omwordri  7239  omlimcl  7245  oaabs2  7312