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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.
StepHypRef Expression
1 oafnex 6109 . . . . 5  |-  ( x  e.  _V  |->  suc  x
)  Fn  _V
21a1i 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
)
72, 3, 4, 5, 6rdgss 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 )
)
83, 4jca 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
) )
108, 9syl 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 ) )
113, 5jca 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
) )
1311, 12syl 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 ) )
147, 10, 133sstr4d 3051 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  +o  A )  C_  ( C  +o  B ) )
1514ex 113 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  ( A  C_  B  ->  ( C  +o  A )  C_  ( C  +o  B
) ) )
Colors of variables: wff set class
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
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