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Theorem onmindif 5522
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 3443 . . . 4  |-  ( x  e.  ( A  \  suc  B )  <->  ( x  e.  A  /\  -.  x  e.  suc  B ) )
2 ssel2 3456 . . . . . . . . 9  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
3 ontri1 5467 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  -.  B  e.  x ) )
4 onsssuc 5520 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  x  e.  suc  B ) )
53, 4bitr3d 258 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  x  <->  x  e.  suc  B ) )
65con1bid 331 . . . . . . . . 9  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( -.  x  e. 
suc  B  <->  B  e.  x
) )
72, 6sylan 473 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  B  e.  On )  ->  ( -.  x  e.  suc  B  <->  B  e.  x ) )
87biimpd 210 . . . . . . 7  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  B  e.  On )  ->  ( -.  x  e.  suc  B  ->  B  e.  x ) )
98exp31 607 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  ( B  e.  On  ->  ( -.  x  e.  suc  B  ->  B  e.  x
) ) ) )
109com23 81 . . . . 5  |-  ( A 
C_  On  ->  ( B  e.  On  ->  (
x  e.  A  -> 
( -.  x  e. 
suc  B  ->  B  e.  x ) ) ) )
1110imp4b 593 . . . 4  |-  ( ( A  C_  On  /\  B  e.  On )  ->  (
( x  e.  A  /\  -.  x  e.  suc  B )  ->  B  e.  x ) )
121, 11syl5bi 220 . . 3  |-  ( ( A  C_  On  /\  B  e.  On )  ->  (
x  e.  ( A 
\  suc  B )  ->  B  e.  x ) )
1312ralrimiv 2835 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  A. x  e.  ( A  \  suc  B ) B  e.  x
)
14 elintg 4257 . . 3  |-  ( B  e.  On  ->  ( B  e.  |^| ( A 
\  suc  B )  <->  A. x  e.  ( A 
\  suc  B ) B  e.  x )
)
1514adantl 467 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  ( B  e.  |^| ( A 
\  suc  B )  <->  A. x  e.  ( A 
\  suc  B ) B  e.  x )
)
1613, 15mpbird 235 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 187    /\ wa 370    e. wcel 1867   A.wral 2773    \ cdif 3430    C_ wss 3433   |^|cint 4249   Oncon0 5433   suc csuc 5435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-ord 5436  df-on 5437  df-suc 5439
This theorem is referenced by:  unblem3  7822  fin23lem26  8744
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