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Theorem oneqmini 4443
Description: A way to show that an ordinal number equals the minimum of a 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
oneqmini  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B
) )
Distinct variable groups:    x, A    x, B

Proof of Theorem oneqmini
StepHypRef Expression
1 ssint 3878 . . . . . 6  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
2 ssel 3174 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( A  e.  B  ->  A  e.  On ) )
3 ssel 3174 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( x  e.  B  ->  x  e.  On ) )
42, 3anim12d 546 . . . . . . . . . . 11  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  e.  On  /\  x  e.  On ) ) )
5 ontri1 4426 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  C_  x  <->  -.  x  e.  A ) )
64, 5syl6 29 . . . . . . . . . 10  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  C_  x  <->  -.  x  e.  A ) ) )
76expdimp 426 . . . . . . . . 9  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
x  e.  B  -> 
( A  C_  x  <->  -.  x  e.  A ) ) )
87pm5.74d 238 . . . . . . . 8  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  B  ->  -.  x  e.  A
) ) )
9 con2b 324 . . . . . . . 8  |-  ( ( x  e.  B  ->  -.  x  e.  A
)  <->  ( x  e.  A  ->  -.  x  e.  B ) )
108, 9syl6bb 252 . . . . . . 7  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  A  ->  -.  x  e.  B
) ) )
1110ralbidv2 2565 . . . . . 6  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  B  A  C_  x  <->  A. x  e.  A  -.  x  e.  B ) )
121, 11syl5bb 248 . . . . 5  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A  C_  |^| B  <->  A. x  e.  A  -.  x  e.  B ) )
1312biimprd 214 . . . 4  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  A  -.  x  e.  B  ->  A  C_  |^| B ) )
1413expimpd 586 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  C_  |^| B ) )
15 intss1 3877 . . . . 5  |-  ( A  e.  B  ->  |^| B  C_  A )
1615a1i 10 . . . 4  |-  ( B 
C_  On  ->  ( A  e.  B  ->  |^| B  C_  A ) )
1716adantrd 454 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  |^| B  C_  A
) )
1814, 17jcad 519 . 2  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  ( A  C_  |^| B  /\  |^| B  C_  A
) ) )
19 eqss 3194 . 2  |-  ( A  =  |^| B  <->  ( A  C_ 
|^| B  /\  |^| B  C_  A ) )
2018, 19syl6ibr 218 1  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  =  |^| B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   |^|cint 3862   Oncon0 4392
This theorem is referenced by:  oneqmin  4596  alephval3  7737  cfsuc  7883  alephval2  8194
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-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
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-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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