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Theorem onmindif2 6642
Description: The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed. (Contributed by NM, 20-Nov-2003.)
Assertion
Ref Expression
onmindif2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  |^| ( A  \  { |^| A } ) )

Proof of Theorem onmindif2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldifsn 4158 . . . 4  |-  ( x  e.  ( A  \  { |^| A } )  <-> 
( x  e.  A  /\  x  =/=  |^| A
) )
2 onnmin 6633 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  x  e.  A )  ->  -.  x  e.  |^| A )
32adantlr 714 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  x  e.  |^| A )
4 oninton 6630 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
54adantr 465 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  |^| A  e.  On )
6 ssel2 3504 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
76adantlr 714 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  On )
8 ontri1 4918 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  -.  x  e.  |^| A ) )
9 onsseleq 4925 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
108, 9bitr3d 255 . . . . . . . . . 10  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( -.  x  e.  |^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
115, 7, 10syl2anc 661 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  x  e. 
|^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
123, 11mpbid 210 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( |^| A  e.  x  \/  |^| A  =  x ) )
1312ord 377 . . . . . . 7  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  |^| A  =  x ) )
14 eqcom 2476 . . . . . . 7  |-  ( |^| A  =  x  <->  x  =  |^| A )
1513, 14syl6ib 226 . . . . . 6  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  x  = 
|^| A ) )
1615necon1ad 2683 . . . . 5  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( x  =/=  |^| A  ->  |^| A  e.  x
) )
1716expimpd 603 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
( x  e.  A  /\  x  =/=  |^| A
)  ->  |^| A  e.  x ) )
181, 17syl5bi 217 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
x  e.  ( A 
\  { |^| A } )  ->  |^| A  e.  x ) )
1918ralrimiv 2879 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
20 intex 4609 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21 elintg 4296 . . . 4  |-  ( |^| A  e.  _V  ->  (
|^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2220, 21sylbi 195 . . 3  |-  ( A  =/=  (/)  ->  ( |^| A  e.  |^| ( A 
\  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2322adantl 466 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  ( |^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2419, 23mpbird 232 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  |^| ( A  \  { |^| A } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    \ cdif 3478    C_ wss 3481   (/)c0 3790   {csn 4033   |^|cint 4288   Oncon0 4884
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888
This theorem is referenced by: (None)
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