Theorem oawordi 6049

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 6024	. . . . 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 909	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> C e. On )
4		simpl1 907	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A e. On )
5		simpl2 908	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> B e. On )
6		simpr 103	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A C_ B )
7	2, 3, 4, 5, 6	rdgss 5970	. . 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 290	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ A e. On ) )
9		oav 6034	. . . 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 290	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ B e. On ) )
12		oav 6034	. . . 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 2988	. 2 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C +o A ) C_ ( C +o B ) )
15	14	ex 108	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 97   /\ w3a 885   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   |-> cmpt 3818   Oncon0 4100   suc csuc 4102   Fn wfn 4897   ` cfv 4902  (class class class)co 5512   reccrdg 5956   +o coa 5998

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-irdg 5957  df-oadd 6005

This theorem is referenced by:  oaword1  6050