Theorem oawordi 6134

Description: Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.)

Assertion

Ref	Expression
oawordi	|- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B -> ( C +o A ) C_ ( C +o B ) ) )

Proof of Theorem oawordi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oafnex 6109	. . . . 5 |- ( x e. _V |-> suc x ) Fn _V
2	1	a1i 9	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( x e. _V |-> suc x ) Fn _V )
3		simpl3 944	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> C e. On )
4		simpl1 942	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A e. On )
5		simpl2 943	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> B e. On )
6		simpr 108	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A C_ B )
7	2, 3, 4, 5, 6	rdgss 6053	. . 3 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( rec ( ( x e. _V |-> suc x ) , C ) ` A ) C_ ( rec ( ( x e. _V |-> suc x ) , C ) ` B ) )
8	3, 4	jca 300	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ A e. On ) )
9		oav 6119	. . . 4 |- ( ( C e. On /\ A e. On ) -> ( C +o A ) = ( rec ( ( x e. _V |-> suc x ) , C ) ` A ) )
10	8, 9	syl 14	. . 3 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C +o A ) = ( rec ( ( x e. _V |-> suc x ) , C ) ` A ) )
11	3, 5	jca 300	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ B e. On ) )
12		oav 6119	. . . 4 |- ( ( C e. On /\ B e. On ) -> ( C +o B ) = ( rec ( ( x e. _V |-> suc x ) , C ) ` B ) )
13	11, 12	syl 14	. . 3 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C +o B ) = ( rec ( ( x e. _V |-> suc x ) , C ) ` B ) )
14	7, 10, 13	3sstr4d 3051	. 2 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C +o A ) C_ ( C +o B ) )
15	14	ex 113	1 |- ( ( A e. On /\ B e. On /\ C e. On ) -> ( A C_ B -> ( C +o A ) C_ ( C +o B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   = wceq 1285   e. wcel 1434   _Vcvv 2610   C_ wss 2982   |-> cmpt 3859   Oncon0 4146   suc csuc 4148   Fn wfn 4947   ` cfv 4952  (class class class)co 5564   reccrdg 6039   +o coa 6083

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-recs 5975  df-irdg 6040  df-oadd 6090

This theorem is referenced by:  oaword1  6136