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Theorem oneqmini 4624
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 4058 . . . . . 6  |-  ( A 
C_  |^| B  <->  A. x  e.  B  A  C_  x
)
2 ssel 3334 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( A  e.  B  ->  A  e.  On ) )
3 ssel 3334 . . . . . . . . . . . 12  |-  ( B 
C_  On  ->  ( x  e.  B  ->  x  e.  On ) )
42, 3anim12d 547 . . . . . . . . . . 11  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  e.  On  /\  x  e.  On ) ) )
5 ontri1 4607 . . . . . . . . . . 11  |-  ( ( A  e.  On  /\  x  e.  On )  ->  ( A  C_  x  <->  -.  x  e.  A ) )
64, 5syl6 31 . . . . . . . . . 10  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  x  e.  B )  ->  ( A  C_  x  <->  -.  x  e.  A ) ) )
76expdimp 427 . . . . . . . . 9  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
x  e.  B  -> 
( A  C_  x  <->  -.  x  e.  A ) ) )
87pm5.74d 239 . . . . . . . 8  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  B  ->  -.  x  e.  A
) ) )
9 con2b 325 . . . . . . . 8  |-  ( ( x  e.  B  ->  -.  x  e.  A
)  <->  ( x  e.  A  ->  -.  x  e.  B ) )
108, 9syl6bb 253 . . . . . . 7  |-  ( ( B  C_  On  /\  A  e.  B )  ->  (
( x  e.  B  ->  A  C_  x )  <->  ( x  e.  A  ->  -.  x  e.  B
) ) )
1110ralbidv2 2719 . . . . . 6  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  B  A  C_  x  <->  A. x  e.  A  -.  x  e.  B ) )
121, 11syl5bb 249 . . . . 5  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A  C_  |^| B  <->  A. x  e.  A  -.  x  e.  B ) )
1312biimprd 215 . . . 4  |-  ( ( B  C_  On  /\  A  e.  B )  ->  ( A. x  e.  A  -.  x  e.  B  ->  A  C_  |^| B ) )
1413expimpd 587 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  A  C_  |^| B ) )
15 intss1 4057 . . . . 5  |-  ( A  e.  B  ->  |^| B  C_  A )
1615a1i 11 . . . 4  |-  ( B 
C_  On  ->  ( A  e.  B  ->  |^| B  C_  A ) )
1716adantrd 455 . . 3  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  |^| B  C_  A
) )
1814, 17jcad 520 . 2  |-  ( B 
C_  On  ->  ( ( A  e.  B  /\  A. x  e.  A  -.  x  e.  B )  ->  ( A  C_  |^| B  /\  |^| B  C_  A
) ) )
19 eqss 3355 . 2  |-  ( A  =  |^| B  <->  ( A  C_ 
|^| B  /\  |^| B  C_  A ) )
2018, 19syl6ibr 219 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   |^|cint 4042   Oncon0 4573
This theorem is referenced by:  oneqmin  4777  alephval3  7983  cfsuc  8129  alephval2  8439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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