Theorem oawordi 6536

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 6511	. . . . 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 1004	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> C e. On )
4		simpl1 1002	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A e. On )
5		simpl2 1003	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> B e. On )
6		simpr 110	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A C_ B )
7	2, 3, 4, 5, 6	rdgss 6450	. . 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 306	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ A e. On ) )
9		oav 6521	. . . 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 306	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C e. On /\ B e. On ) )
12		oav 6521	. . . 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 3229	. 2 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> ( C +o A ) C_ ( C +o B ) )
15	14	ex 115	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 104   /\ w3a 980   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   |-> cmpt 4095   Oncon0 4399   suc csuc 4401   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   reccrdg 6436   +o coa 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-recs 6372  df-irdg 6437  df-oadd 6487

This theorem is referenced by:  oaword1  6538