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Theorem onmindif 4813
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 3343 . . . 4  |-  ( x  e.  ( A  \  suc  B )  <->  ( x  e.  A  /\  -.  x  e.  suc  B ) )
2 ssel2 3356 . . . . . . . . 9  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
3 ontri1 4758 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  B  e.  On )  ->  ( x  C_  B  <->  -.  B  e.  x ) )
4 onsssuc 4811 . . . . . . . . . . 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 2803 . 2  |-  ( ( A  C_  On  /\  B  e.  On )  ->  A. x  e.  ( A  \  suc  B ) B  e.  x
)
14 elintg 4141 . . 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 1756   A.wral 2720    \ cdif 3330    C_ wss 3333   |^|cint 4133   Oncon0 4724   suc csuc 4726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-int 4134  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-suc 4730
This theorem is referenced by:  unblem3  7571  fin23lem26  8499
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