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Theorem oneqmin 6639
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 6629 . . . 4  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  |^| B  e.  B )
2 eleq1 2529 . . . 4  |-  ( A  =  |^| B  -> 
( A  e.  B  <->  |^| B  e.  B ) )
31, 2syl5ibrcom 222 . . 3  |-  ( ( B  C_  On  /\  B  =/=  (/) )  ->  ( A  =  |^| B  ->  A  e.  B )
)
4 eleq2 2530 . . . . . . 7  |-  ( A  =  |^| B  -> 
( x  e.  A  <->  x  e.  |^| B ) )
54biimpd 207 . . . . . 6  |-  ( A  =  |^| B  -> 
( x  e.  A  ->  x  e.  |^| B
) )
6 onnmin 6637 . . . . . . . 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 2873 . . . 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 4938 . . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    C_ wss 3471   (/)c0 3793   |^|cint 4288   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by: (None)
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