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Theorem onmindif 4967
Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)
Assertion
Ref Expression
onmindif  |-  ( ( A  C_  On  /\  B  e.  On )  ->  B  e.  |^| ( A  \  suc  B ) )

Proof of Theorem onmindif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldif 3486 . . . 4  |-  ( x  e.  ( A  \  suc  B )  <->  ( x  e.  A  /\  -.  x  e.  suc  B ) )
2 ssel2 3499 . . . . . . . . 9  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
3 ontri1 4912 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  -.  B  e.  x ) )
4 onsssuc 4965 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  x  e.  suc  B ) )
53, 4bitr3d 255 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  x  <->  x  e.  suc  B ) )
65con1bid 330 . . . . . . . . 9  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( -.  x  e. 
suc  B  <->  B  e.  x
) )
72, 6sylan 471 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  B  e.  On )  ->  ( -.  x  e.  suc  B  <->  B  e.  x ) )
87biimpd 207 . . . . . . 7  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  B  e.  On )  ->  ( -.  x  e.  suc  B  ->  B  e.  x ) )
98exp31 604 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  ( B  e.  On  ->  ( -.  x  e.  suc  B  ->  B  e.  x
) ) ) )
109com23 78 . . . . 5  |-  ( A 
C_  On  ->  ( B  e.  On  ->  (
x  e.  A  -> 
( -.  x  e. 
suc  B  ->  B  e.  x ) ) ) )
1110imp4b 590 . . . 4  |-  ( ( A  C_  On  /\  B  e.  On )  ->  (
( x  e.  A  /\  -.  x  e.  suc  B )  ->  B  e.  x ) )
121, 11syl5bi 217 . . 3  |-  ( ( A  C_  On  /\  B  e.  On )  ->  (
x  e.  ( A 
\  suc  B )  ->  B  e.  x ) )
1312ralrimiv 2876 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  A. x  e.  ( A  \  suc  B ) B  e.  x
)
14 elintg 4290 . . 3  |-  ( B  e.  On  ->  ( B  e.  |^| ( A 
\  suc  B )  <->  A. x  e.  ( A 
\  suc  B ) B  e.  x )
)
1514adantl 466 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  ( B  e.  |^| ( A 
\  suc  B )  <->  A. x  e.  ( A 
\  suc  B ) B  e.  x )
)
1613, 15mpbird 232 1  |-  ( ( A  C_  On  /\  B  e.  On )  ->  B  e.  |^| ( A  \  suc  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   A.wral 2814    \ cdif 3473    C_ wss 3476   |^|cint 4282   Oncon0 4878   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884
This theorem is referenced by:  unblem3  7774  fin23lem26  8705
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