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Theorem onmindif2 4792
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 3927 . . . 4  |-  ( x  e.  ( A  \  { |^| A } )  <-> 
( x  e.  A  /\  x  =/=  |^| A
) )
2 onnmin 4783 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  x  e.  A )  ->  -.  x  e.  |^| A )
32adantlr 696 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  x  e.  |^| A )
4 oninton 4780 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
54adantr 452 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  |^| A  e.  On )
6 ssel2 3343 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
76adantlr 696 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  On )
8 ontri1 4615 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  -.  x  e.  |^| A ) )
9 onsseleq 4622 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
108, 9bitr3d 247 . . . . . . . . . 10  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( -.  x  e.  |^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
115, 7, 10syl2anc 643 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  x  e. 
|^| A  <->  ( |^| A  e.  x  \/  |^| A  =  x ) ) )
123, 11mpbid 202 . . . . . . . 8  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( |^| A  e.  x  \/  |^| A  =  x ) )
1312ord 367 . . . . . . 7  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  |^| A  =  x ) )
14 eqcom 2438 . . . . . . 7  |-  ( |^| A  =  x  <->  x  =  |^| A )
1513, 14syl6ib 218 . . . . . 6  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  x  = 
|^| A ) )
1615necon1ad 2671 . . . . 5  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( x  =/=  |^| A  ->  |^| A  e.  x
) )
1716expimpd 587 . . . 4  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
( x  e.  A  /\  x  =/=  |^| A
)  ->  |^| A  e.  x ) )
181, 17syl5bi 209 . . 3  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  (
x  e.  ( A 
\  { |^| A } )  ->  |^| A  e.  x ) )
1918ralrimiv 2788 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
20 intex 4356 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21 elintg 4058 . . . 4  |-  ( |^| A  e.  _V  ->  (
|^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2220, 21sylbi 188 . . 3  |-  ( A  =/=  (/)  ->  ( |^| A  e.  |^| ( A 
\  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2322adantl 453 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  ( |^| A  e.  |^| ( A  \  { |^| A } )  <->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
)
2419, 23mpbird 224 1  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  |^| ( A  \  { |^| A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   _Vcvv 2956    \ cdif 3317    C_ wss 3320   (/)c0 3628   {csn 3814   |^|cint 4050   Oncon0 4581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585
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