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Theorem onmindif 4482
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 3162 . . . 4  |-  ( x  e.  ( A  \  suc  B )  <->  ( x  e.  A  /\  -.  x  e.  suc  B ) )
2 ssel2 3175 . . . . . . . . 9  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
3 ontri1 4426 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  -.  B  e.  x ) )
4 onsssuc 4480 . . . . . . . . . . 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 2625 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  A. x  e.  ( A  \  suc  B ) B  e.  x
)
14 elintg 3870 . . 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 1684   A.wral 2543    \ cdif 3149    C_ wss 3152   |^|cint 3862   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  unblem3  7111  fin23lem26  7951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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