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Theorem oawordi 6244
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 6219 . . . . 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 949 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  C  e.  On )
4 simpl1 947 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  A  e.  On )
5 simpl2 948 . . . 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
)
72, 3, 4, 5, 6rdgss 6162 . . 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 301 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  e.  On  /\  A  e.  On ) )
9 oav 6229 . . . 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 301 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  e.  On  /\  B  e.  On ) )
12 oav 6229 . . . 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 3070 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  +o  A )  C_  ( C  +o  B ) )
1514ex 114 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 103    /\ w3a 925    = wceq 1290    e. wcel 1439   _Vcvv 2620    C_ wss 3000    |-> cmpt 3905   Oncon0 4199   suc csuc 4201    Fn wfn 5023   ` cfv 5028  (class class class)co 5666   reccrdg 6148    +o coa 6192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-irdg 6149  df-oadd 6199
This theorem is referenced by:  oaword1  6246
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