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Theorem onmindif 4498
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 3175 . . . 4  |-  ( x  e.  ( A  \  suc  B )  <->  ( x  e.  A  /\  -.  x  e.  suc  B ) )
2 ssel2 3188 . . . . . . . . 9  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
3 ontri1 4442 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  -.  B  e.  x ) )
4 onsssuc 4496 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  x  e.  suc  B ) )
53, 4bitr3d 246 . . . . . . . . . 10  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( -.  B  e.  x  <->  x  e.  suc  B ) )
65con1bid 320 . . . . . . . . 9  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( -.  x  e. 
suc  B  <->  B  e.  x
) )
72, 6sylan 457 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  B  e.  On )  ->  ( -.  x  e.  suc  B  <->  B  e.  x ) )
87biimpd 198 . . . . . . 7  |-  ( ( ( A  C_  On  /\  x  e.  A )  /\  B  e.  On )  ->  ( -.  x  e.  suc  B  ->  B  e.  x ) )
98exp31 587 . . . . . 6  |-  ( A 
C_  On  ->  ( x  e.  A  ->  ( B  e.  On  ->  ( -.  x  e.  suc  B  ->  B  e.  x
) ) ) )
109com23 72 . . . . 5  |-  ( A 
C_  On  ->  ( B  e.  On  ->  (
x  e.  A  -> 
( -.  x  e. 
suc  B  ->  B  e.  x ) ) ) )
1110imp4b 573 . . . 4  |-  ( ( A  C_  On  /\  B  e.  On )  ->  (
( x  e.  A  /\  -.  x  e.  suc  B )  ->  B  e.  x ) )
121, 11syl5bi 208 . . 3  |-  ( ( A  C_  On  /\  B  e.  On )  ->  (
x  e.  ( A 
\  suc  B )  ->  B  e.  x ) )
1312ralrimiv 2638 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  A. x  e.  ( A  \  suc  B ) B  e.  x
)
14 elintg 3886 . . 3  |-  ( B  e.  On  ->  ( B  e.  |^| ( A 
\  suc  B )  <->  A. x  e.  ( A 
\  suc  B ) B  e.  x )
)
1514adantl 452 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  ( B  e.  |^| ( A 
\  suc  B )  <->  A. x  e.  ( A 
\  suc  B ) B  e.  x )
)
1613, 15mpbird 223 1  |-  ( ( A  C_  On  /\  B  e.  On )  ->  B  e.  |^| ( A  \  suc  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   A.wral 2556    \ cdif 3162    C_ wss 3165   |^|cint 3878   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  unblem3  7127  fin23lem26  7967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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