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Theorem onmindif2 6546
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 4069 . . . 4  |-  ( x  e.  ( A  \  { |^| A } )  <-> 
( x  e.  A  /\  x  =/=  |^| A
) )
2 onnmin 6537 . . . . . . . . . 10  |-  ( ( A  C_  On  /\  x  e.  A )  ->  -.  x  e.  |^| A )
32adantlr 712 . . . . . . . . 9  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  x  e.  |^| A )
4 oninton 6534 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  |^| A  e.  On )
54adantr 463 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  |^| A  e.  On )
6 ssel2 3412 . . . . . . . . . . 11  |-  ( ( A  C_  On  /\  x  e.  A )  ->  x  e.  On )
76adantlr 712 . . . . . . . . . 10  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  x  e.  On )
8 ontri1 4826 . . . . . . . . . . 11  |-  ( (
|^| A  e.  On  /\  x  e.  On )  ->  ( |^| A  C_  x  <->  -.  x  e.  |^| A ) )
9 onsseleq 4833 . . . . . . . . . . 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 659 . . . . . . . . 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 375 . . . . . . 7  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  |^| A  =  x ) )
14 eqcom 2391 . . . . . . 7  |-  ( |^| A  =  x  <->  x  =  |^| A )
1513, 14syl6ib 226 . . . . . 6  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( -.  |^| A  e.  x  ->  x  = 
|^| A ) )
1615necon1ad 2598 . . . . 5  |-  ( ( ( A  C_  On  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( x  =/=  |^| A  ->  |^| A  e.  x
) )
1716expimpd 601 . . . 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 2794 . 2  |-  ( ( A  C_  On  /\  A  =/=  (/) )  ->  A. x  e.  ( A  \  { |^| A } ) |^| A  e.  x )
20 intex 4521 . . . 4  |-  ( A  =/=  (/)  <->  |^| A  e.  _V )
21 elintg 4207 . . . 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 464 . 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 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   _Vcvv 3034    \ cdif 3386    C_ wss 3389   (/)c0 3711   {csn 3944   |^|cint 4199   Oncon0 4792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796
This theorem is referenced by: (None)
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