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Theorem onmindif2 4603
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 3749 . . . 4  |-  ( x  e.  ( A  \  { |^| A } )  <-> 
( x  e.  A  /\  x  =/=  |^| A
) )
2 onnmin 4594 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  x  e.  A )  ->  -.  x  e.  |^| A )
32adantlr 695 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  x  e.  |^| A )
4 oninton 4591 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
54adantr 451 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  |^| A  e.  On )
6 ssel2 3175 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
76adantlr 695 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  On )
8 ontri1 4426 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  -.  x  e.  |^| A ) )
9 onsseleq 4433 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
108, 9bitr3d 246 . . . . . . . . . 10  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( -.  x  e.  |^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
115, 7, 10syl2anc 642 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  x  e. 
|^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
123, 11mpbid 201 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( |^| A  e.  x  \/  |^| A  =  x ) )
1312ord 366 . . . . . . 7  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  |^| A  =  x ) )
14 eqcom 2285 . . . . . . 7  |-  ( |^| A  =  x  <->  x  =  |^| A )
1513, 14syl6ib 217 . . . . . 6  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  x  = 
|^| A ) )
1615necon1ad 2513 . . . . 5  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( x  =/=  |^| A  ->  |^| A  e.  x
) )
1716expimpd 586 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
( x  e.  A  /\  x  =/=  |^| A
)  ->  |^| A  e.  x ) )
181, 17syl5bi 208 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
x  e.  ( A 
\  { |^| A } )  ->  |^| A  e.  x ) )
1918ralrimiv 2625 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
20 intex 4167 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21 elintg 3870 . . . 4  |-  ( |^| A  e.  _V  ->  (
|^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2220, 21sylbi 187 . . 3  |-  ( A  =/=  (/)  ->  ( |^| A  e.  |^| ( A 
\  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2322adantl 452 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  ( |^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2419, 23mpbird 223 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  |^| ( A  \  { |^| A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   |^|cint 3862   Oncon0 4392
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-13 1686  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  ax-un 4512
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-tp 3648  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
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