Theorem oawordi 6373

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 6348	. . . . 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 987	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> C e. On )
4		simpl1 985	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A e. On )
5		simpl2 986	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> B e. On )
6		simpr 109	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A C_ B )
7	2, 3, 4, 5, 6	rdgss 6288	. . 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 304	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ A e. On ) )
9		oav 6358	. . . 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 304	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ B e. On ) )
12		oav 6358	. . . 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 3147	. 2 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C +o A ) C_ ( C +o B ) )
15	14	ex 114	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 103   /\ w3a 963   = wceq 1332   e. wcel 1481   _Vcvv 2689   C_ wss 3076   |-> cmpt 3997   Oncon0 4293   suc csuc 4295   Fn wfn 5126   ` cfv 5131  (class class class)co 5782   reccrdg 6274   +o coa 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-recs 6210  df-irdg 6275  df-oadd 6325

This theorem is referenced by:  oaword1  6375