Theorem oawordi 6704

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 6679	. . . . 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 1029	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> C e. On )
4		simpl1 1027	. . . 4 |- ( ( ( A e. On /\ B e. On /\ C e. On ) /\ A C_ B ) -> A e. On )
5		simpl2 1028	. . . 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 6616	. . 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 6689	. . . 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 6689	. . . 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 3285	. 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 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213   |-> cmpt 4173   Oncon0 4486   suc csuc 4488   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   reccrdg 6602   +o coa 6646

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-recs 6538  df-irdg 6603  df-oadd 6653

This theorem is referenced by:  oaword1  6706