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Theorem oneqmin 4718
Description: A way to show that an ordinal number equals the minimum of a non-empty collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)
Assertion
Ref Expression
oneqmin  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  <->  ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B ) ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem oneqmin
StepHypRef Expression
1 onint 4708 . . . 4  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  |^| B  e.  B )
2 eleq1 2440 . . . 4  |-  ( A  =  |^| B  -> 
( A  e.  B  <->  |^| B  e.  B ) )
31, 2syl5ibrcom 214 . . 3  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  ->  A  e.  B )
)
4 eleq2 2441 . . . . . . 7  |-  ( A  =  |^| B  -> 
( x  e.  A  <->  x  e.  |^| B ) )
54biimpd 199 . . . . . 6  |-  ( A  =  |^| B  -> 
( x  e.  A  ->  x  e.  |^| B
) )
6 onnmin 4716 . . . . . . . 8  |-  ( ( B  C_  On  /\  x  e.  B )  ->  -.  x  e.  |^| B )
76ex 424 . . . . . . 7  |-  ( B 
C_  On  ->  ( x  e.  B  ->  -.  x  e.  |^| B ) )
87con2d 109 . . . . . 6  |-  ( B 
C_  On  ->  ( x  e.  |^| B  ->  -.  x  e.  B )
)
95, 8syl9r 69 . . . . 5  |-  ( B 
C_  On  ->  ( A  =  |^| B  -> 
( x  e.  A  ->  -.  x  e.  B
) ) )
109ralrimdv 2731 . . . 4  |-  ( B 
C_  On  ->  ( A  =  |^| B  ->  A. x  e.  A  -.  x  e.  B
) )
1110adantr 452 . . 3  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  ->  A. x  e.  A  -.  x  e.  B
) )
123, 11jcad 520 . 2  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  -> 
( A  e.  B  /\  A. x  e.  A  -.  x  e.  B
) ) )
13 oneqmini 4566 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B
) )
1413adantr 452 . 2  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  (
( A  e.  B  /\  A. x  e.  A  -.  x  e.  B
)  ->  A  =  |^| B ) )
1512, 14impbid 184 1  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  <->  ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642    C_ wss 3256   (/)c0 3564   |^|cint 3985   Oncon0 4515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-br 4147  df-opab 4201  df-tr 4237  df-eprel 4428  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519
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