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Theorem oneqmin 6618
Description: A way to show that an ordinal number equals the minimum of a nonempty 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 6608 . . . 4  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  |^| B  e.  B )
2 eleq1 2539 . . . 4  |-  ( A  =  |^| B  -> 
( A  e.  B  <->  |^| B  e.  B ) )
31, 2syl5ibrcom 222 . . 3  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  ->  A  e.  B )
)
4 eleq2 2540 . . . . . . 7  |-  ( A  =  |^| B  -> 
( x  e.  A  <->  x  e.  |^| B ) )
54biimpd 207 . . . . . 6  |-  ( A  =  |^| B  -> 
( x  e.  A  ->  x  e.  |^| B
) )
6 onnmin 6616 . . . . . . . 8  |-  ( ( B  C_  On  /\  x  e.  B )  ->  -.  x  e.  |^| B )
76ex 434 . . . . . . 7  |-  ( B 
C_  On  ->  ( x  e.  B  ->  -.  x  e.  |^| B ) )
87con2d 115 . . . . . 6  |-  ( B 
C_  On  ->  ( x  e.  |^| B  ->  -.  x  e.  B )
)
95, 8syl9r 72 . . . . 5  |-  ( B 
C_  On  ->  ( A  =  |^| B  -> 
( x  e.  A  ->  -.  x  e.  B
) ) )
109ralrimdv 2880 . . . 4  |-  ( B 
C_  On  ->  ( A  =  |^| B  ->  A. x  e.  A  -.  x  e.  B
) )
1110adantr 465 . . 3  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  ->  A. x  e.  A  -.  x  e.  B
) )
123, 11jcad 533 . 2  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  -> 
( A  e.  B  /\  A. x  e.  A  -.  x  e.  B
) ) )
13 oneqmini 4929 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B
) )
1413adantr 465 . 2  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  (
( A  e.  B  /\  A. x  e.  A  -.  x  e.  B
)  ->  A  =  |^| B ) )
1512, 14impbid 191 1  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  <->  ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    C_ wss 3476   (/)c0 3785   |^|cint 4282   Oncon0 4878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882
This theorem is referenced by: (None)
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