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Theorem onmindif2 6658
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 4088 . . . 4  |-  ( x  e.  ( A  \  { |^| A } )  <-> 
( x  e.  A  /\  x  =/=  |^| A
) )
2 onnmin 6649 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  x  e.  A )  ->  -.  x  e.  |^| A )
32adantlr 729 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  x  e.  |^| A )
4 oninton 6646 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
54adantr 472 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  |^| A  e.  On )
6 ssel2 3413 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
76adantlr 729 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  On )
8 ontri1 5464 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  -.  x  e.  |^| A ) )
9 onsseleq 5471 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
108, 9bitr3d 263 . . . . . . . . . 10  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( -.  x  e.  |^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
115, 7, 10syl2anc 673 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  x  e. 
|^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
123, 11mpbid 215 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( |^| A  e.  x  \/  |^| A  =  x ) )
1312ord 384 . . . . . . 7  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  |^| A  =  x ) )
14 eqcom 2478 . . . . . . 7  |-  ( |^| A  =  x  <->  x  =  |^| A )
1513, 14syl6ib 234 . . . . . 6  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  x  = 
|^| A ) )
1615necon1ad 2660 . . . . 5  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( x  =/=  |^| A  ->  |^| A  e.  x
) )
1716expimpd 614 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
( x  e.  A  /\  x  =/=  |^| A
)  ->  |^| A  e.  x ) )
181, 17syl5bi 225 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
x  e.  ( A 
\  { |^| A } )  ->  |^| A  e.  x ) )
1918ralrimiv 2808 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
20 intex 4557 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21 elintg 4234 . . . 4  |-  ( |^| A  e.  _V  ->  (
|^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2220, 21sylbi 200 . . 3  |-  ( A  =/=  (/)  ->  ( |^| A  e.  |^| ( A 
\  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2322adantl 473 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  ( |^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2419, 23mpbird 240 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  |^| ( A  \  { |^| A } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   {csn 3959   |^|cint 4226   Oncon0 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434
This theorem is referenced by: (None)
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