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Theorem oneqmin 4596
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 4586 . . . 4  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  |^| B  e.  B )
2 eleq1 2343 . . . 4  |-  ( A  =  |^| B  -> 
( A  e.  B  <->  |^| B  e.  B ) )
31, 2syl5ibrcom 213 . . 3  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  ->  A  e.  B )
)
4 eleq2 2344 . . . . . . 7  |-  ( A  =  |^| B  -> 
( x  e.  A  <->  x  e.  |^| B ) )
54biimpd 198 . . . . . 6  |-  ( A  =  |^| B  -> 
( x  e.  A  ->  x  e.  |^| B
) )
6 onnmin 4594 . . . . . . . 8  |-  ( ( B  C_  On  /\  x  e.  B )  ->  -.  x  e.  |^| B )
76ex 423 . . . . . . 7  |-  ( B 
C_  On  ->  ( x  e.  B  ->  -.  x  e.  |^| B ) )
87con2d 107 . . . . . 6  |-  ( B 
C_  On  ->  ( x  e.  |^| B  ->  -.  x  e.  B )
)
95, 8syl9r 67 . . . . 5  |-  ( B 
C_  On  ->  ( A  =  |^| B  -> 
( x  e.  A  ->  -.  x  e.  B
) ) )
109ralrimdv 2632 . . . 4  |-  ( B 
C_  On  ->  ( A  =  |^| B  ->  A. x  e.  A  -.  x  e.  B
) )
1110adantr 451 . . 3  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  ->  A. x  e.  A  -.  x  e.  B
) )
123, 11jcad 519 . 2  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  -> 
( A  e.  B  /\  A. x  e.  A  -.  x  e.  B
) ) )
13 oneqmini 4443 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B
) )
1413adantr 451 . 2  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  (
( A  e.  B  /\  A. x  e.  A  -.  x  e.  B
)  ->  A  =  |^| B ) )
1512, 14impbid 183 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   |^|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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