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Theorem oawordi 6494
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 6469 . . . . 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 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
)
72, 3, 4, 5, 6rdgss 6408 . . 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 306 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  e.  On  /\  A  e.  On ) )
9 oav 6479 . . . 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 306 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  e.  On  /\  B  e.  On ) )
12 oav 6479 . . . 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 3215 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  +o  A )  C_  ( C  +o  B ) )
1514ex 115 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160   _Vcvv 2752    C_ wss 3144    |-> cmpt 4079   Oncon0 4381   suc csuc 4383    Fn wfn 5230   ` cfv 5235  (class class class)co 5896   reccrdg 6394    +o coa 6438
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-recs 6330  df-irdg 6395  df-oadd 6445
This theorem is referenced by:  oaword1  6496
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