ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oawordi Unicode version

Theorem oawordi 6445
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 6420 . . . . 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 997 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  C  e.  On )
4 simpl1 995 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  A  e.  On )
5 simpl2 996 . . . 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 6359 . . 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 304 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  e.  On  /\  A  e.  On ) )
9 oav 6430 . . . 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 304 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  /\  A  C_  B )  ->  ( C  e.  On  /\  B  e.  On ) )
12 oav 6430 . . . 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 3192 . 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 973    = wceq 1348    e. wcel 2141   _Vcvv 2730    C_ wss 3121    |-> cmpt 4048   Oncon0 4346   suc csuc 4348    Fn wfn 5191   ` cfv 5196  (class class class)co 5850   reccrdg 6345    +o coa 6389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-recs 6281  df-irdg 6346  df-oadd 6396
This theorem is referenced by:  oaword1  6447
  Copyright terms: Public domain W3C validator